❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❈â♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦
❯♠ ❡st✉❞♦ ❞♦s ▼♦❞❡❧♦s ❞❛
●❡♦♠❡tr✐❛ ❍✐♣❡r❜ó❧✐❝❛
❏♦sé ▼❡ss✐❛s ▼❛❣❛❧❤ã❡s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡
❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ❲❧❛❞✐♠✐r ❙❡✐①❛s
Magalhães, José Messias
Um estudo dos modelos da geometria hiperbólica / José Messias Magalhães. - Rio Claro, 2015
63 f. : il., figs.
Dissertação (mestrado) - Universidade Estadual Paulista, Instituto de Geociências e Ciências Exatas
Orientador: Wladimir Seixas
1. Geometria não-euclidiana. 2. Fundamentos de geometria. 3. Modelos da geometria hiperbólica. I. Título. 516.9
M188e
❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖
❏♦sé ▼❡ss✐❛s ▼❛❣❛❧❤ã❡s
❯♠ ❡st✉❞♦ ❞♦s ▼♦❞❡❧♦s ❞❛
●❡♦♠❡tr✐❛ ❍✐♣❡r❜ó❧✐❝❛
❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿
Pr♦❢✳ ❉r✳ ❲❧❛❞✐♠✐r ❙❡✐①❛s ❖r✐❡♥t❛❞♦r
Pr♦❢❛✳ ❉r❛✳ ❨✉r✐❦♦ ❨❛♠♦♠♦t♦ ❇❛❧❞✐♥ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ❯❋❙❈❛r
Pr♦❢✳ ❉r✳ ❏♦ã♦ P❡r❡s ❱✐❡✐r❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ❯♥❡s♣
❆❣r❛❞❡❝✐♠❡♥t♦s
Pr✐♠❡✐r❛♠❡♥t❡ ❛ ❉❊❯❙ ❝r✐❛❞♦r ❞♦ ✉♥✐✈❡rs♦ ❡ r❡s♣♦♥sá✈❡❧ ♣♦r t✉❞♦ q✉❡ s♦♠♦s ❡ t✉❞♦ q✉❡ t❡♠♦s❀
❆♦s ♠❡✉s ♣❛✐s ✭✐♥ ♠❡♠♦r✐❛♥✮ ♣❡❧♦s ❝✉✐❞❛❞♦s ❡ ♣♦r t❡r❡♠ ♠❡ ❛❥✉❞❛❞♦ ♥♦s ♣r✐♠❡✐r♦s ♣❛ss♦s❀
➚ ♠✐♥❤❛ ❡s♣♦s❛ q✉❡ s❡♠♣r❡ ❡stá ❛♦ ♠❡✉ ❧❛❞♦✱ ♠❡ ❛❥✉❞❛♥❞♦ ❡ ❢❛③❡♥❞♦ ♦ s❡✉ ♠❡❧❤♦r ♣❛r❛ r❡❛❧✐③❛çã♦ ❞♦s ♠❡✉s s♦♥❤♦s❀
❆♦ Pr♦❢❡ss♦r ❉r✳ ❲❧❛❞✐♠✐r ❙❡✐①❛s✱ ♠❛✐s ❞♦ q✉❡ ♠❡✉ ♣r♦❢❡ss♦r✱ ♠❛✐s ❞♦ q✉❡ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ✉♠ ❣r❛♥❞❡ ❛♠✐❣♦✱ q✉❡ ❝♦♠ t♦❞♦ s❡✉ ♣r♦✜ss✐♦♥❛❧✐s♠♦✱ ❞❡❞✐❝❛çã♦ ❡ ♣❛❝✐ê♥❝✐❛ ❛❥✉❞♦✉✲♠❡ ❞❡ ✉♠❛ ❢♦r♠❛ ❞❡❝✐s✐✈❛ ♥❛ ❝♦♥❞✉çã♦ ❡ ❡①❡❝✉çã♦ ❞❡st❛ ❞✐ss❡rt❛çã♦❀
❆♦ ❝♦r♣♦ ❞♦❝❡♥t❡ ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥❡s♣✱ ❈❛♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ❛♦ Pr♦❢❡ss♦r ❉r✳ ❏♦ã♦ P❡r❡s ❱✐❡✐r❛✱ ♣♦r t♦❞♦ ❛♣♦✐♦ ❞✉r❛♥t❡ ♠✐♥❤❛ tr❛❥❡tór✐❛ ♥❡st❛ ❯♥✐✈❡rs✐❞❛❞❡❀
➚ Pr♦❢❡ss♦r❛ ❉r❛✳ ❊❧ír✐s ❈r✐st✐♥❛ ❘✐③③✐♦❧❧✐ ♣♦r s✉❛s s✉❣❡stõ❡s ❡ ❝♦rr❡çõ❡s ♣❛r❛ ❛ ♠❡❧❤♦r✐❛ ❞❡st❡ tr❛❜❛❧❤♦ ♥♦ ❊①❛♠❡ ❞❡ ◗✉❛❧✐✜❝❛çã♦❀
❆♦ ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✱ ♣r♦❢❡ss♦r❡s ❉r❛✳ ❨✉r✐❦♦ ❨❛♠❛♠♦t♦ ❇❛❧❞✐♥ ❡ ❉r✳ ❏♦ã♦ P❡r❡s ❱✐❡✐r❛ ♣❡❧♦s ❝♦♠❡♥tár✐♦s✱ ❡①♣❧✐❝❛çõ❡s✱ s✉❣❡stõ❡s ❡ ❝♦rr❡çõ❡s ♥❛ r❡❞❛çã♦ ✜♥❛❧ ❞❡st❡ tr❛❜❛❧❤♦❀
❆♦ ❈♦r♣♦ ❆❞♠✐♥✐str❛t✐✈♦ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ s❡❝r❡tár✐❛ ❙r❛✳ ■♥❛❥❛r❛ ♣♦r t♦❞♦ ❛♣♦✐♦ ❞❛❞♦ ❞✉r❛♥t❡ ❛ r❡❛❧✐③❛çã♦ ❞♦ ❝✉rs♦ ❞❡ ♠❡str❛❞♦❀
❆♦s ♠❡✉s ✜❧❤♦s ❏ú♥✐♦r✱ ❏♦s❡✈❛♥ ❡ ❏♦❡❧s♦♥ ♣❡❧❛ ♣r❡s❡♥ç❛ t♦❞❛s ❛s ✈❡③❡s q✉❡ ♣r❡❝✐s❡✐❀ ➚s ♠✐♥❤❛s ♥♦r❛s ●✐s❡❧❡ ❡ ❏❛♥❛ ♣♦r t♦❞♦ ❛♣♦✐♦ ❞❛❞♦❀
❆♦s ♠❡✉s ♥❡t✐♥❤♦s ▼❛r✐♥❛✱ ❏♦♥❡s✱ ❏♦s✐❡❧❧❡ ❡ ▲❡tí❝✐❛ q✉❡ ❝♦♠ s✉❛s ✐♥♦❝ê♥❝✐❛s s❡r✈✐✲ r❛♠ ❞❡ ✐♥❝❡♥t✐✈♦ ❡ ❛♣♦✐♦ ❞✉r❛♥t❡ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦❀
❆♦s ♠❡✉s ✐r♠ã♦s✱ ❡♠ ♣❛rt✐❝✉❧❛r à ❏❛❝✐r❛ q✉❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈♦✉ ♥❛ ❜✉s❝❛ ❞❡ ♠❡✉s ♦❜❥❡t✐✈♦s❀
❆ ♠❛t❡♠át✐❝❛✱ q✉❛♥❞♦ ❛ ❝♦♠♣r❡❡♥❞❡♠♦s ❜❡♠✱ ♣♦ss✉✐ ♥ã♦ s♦♠❡♥t❡ ❛ ✈❡r❞❛❞❡✱ ♠❛s t❛♠❜é♠ ❛ s✉♣r❡♠❛ ❜❡❧❡③❛✳
❇❡rtr❛♥❞ ❘✉ss❡❧
◆❛ ♠❛✐♦r ♣❛rt❡ ❞❛s ❝✐ê♥❝✐❛s✱ ✉♠❛ ❣❡r❛çã♦ ♣õ❡ ❛❜❛✐①♦ ♦ q✉❡ ❛ ♦✉tr❛ ❝♦♥str✉✐✉✱ ❡ ♦ q✉❡ ✉♠❛ ❣❡r❛çã♦ ❡st❛❜❡❧❡❝❡✉✱ ❛ ♦✉tr❛ ❞❡s❢❛③✳ ❙♦♠❡♥t❡ ♥❛ ♠❛t❡♠át✐❝❛ é q✉❡ ❝❛❞❛ ❣❡r❛çã♦ ❝♦♥stró✐ ✉♠ ♥♦✈♦ ❛♥❞❛r s♦❜r❡ ❛ ❛♥t✐❣❛ ❡str✉t✉r❛✳
❍❡r♠❛♥♥ ❍❛♥❦❡❧
❯♠❛ ❣❡♦♠❡tr✐❛ ♥ã♦ ♣♦❞❡ s❡r ♠❛✐s ✈❡r❞❛❞❡✐r❛ ❞♦ q✉❡ ❛ ♦✉tr❛❀ ♣♦❞❡rá s❡r ❛♣❡♥❛s ♠❛✐s ❝ô♠♦❞❛✳
❘❡s✉♠♦
❊st❛ ❞✐ss❡rt❛çã♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✐♥tr♦❞✉③✐r ♦s ❝♦♥❝❡✐t♦s ❡ ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ❍✐♣❡r❜ó❧✐❝❛✱ ❡♥tr❡ ❡❧❡s ❛ ♥ã♦ ❡①✐stê♥❝✐❛ ❞❡ r❡tâ♥❣✉❧♦s✳ ❱❡r✐✜❝❛✲s❡ ❛ss✐♠ q✉❡ ❛s ❞✐❢❡r❡♥ç❛s ❡♥tr❡ ❛s ❣❡♦♠❡tr✐❛s ❡✉❝❧✐❞✐❛♥❛ ❡ ❤✐♣❡r❜ó❧✐❝❛ s❡ ❞á ♣❡❧❛ ♥❡❣❛çã♦ ❞♦ ◗✉✐♥t♦ ❆①✐♦♠❛ ❞❡ ❊✉❝❧✐❞❡s ♦✉✱ ❝♦♠♦ é ❝♦♥❤❡❝✐❞♦✱ ♦ ❆①✐♦♠❛ ❞❛s ♣❛r❛❧❡❧❛s ❞❡ ❊✉❝❧✐❞❡s✳ ◆❛ ♣❛rt❡ ✜♥❛❧ ❞❡st❡ tr❛❜❛❧❤♦ ❛❜♦r❞❛r❡♠♦s três ♣r✐♥❝✐♣❛✐s ♠♦❞❡❧♦s ❞❛ ●❡♦♠❡tr✐❛ ❍✐✲ ♣❡r❜ó❧✐❝❛✿ ♦ ❉✐s❝♦ ❞❡ ❇❡❧tr❛♠✐✲❑❧❡✐♥✱ ♦ ❉✐s❝♦ ❞❡ P♦✐♥❝❛ré ❡ ♦ ❙❡♠✐♣❧❛♥♦ ❞❡ P♦✐♥❝❛ré✳ ❉❡♠♦♥str❛r❡♠♦s t❛♠❜é♠ q✉❡ ❡st❡s ♠♦❞❡❧♦s sã♦ ✐s♦♠♦r❢♦s✳
❆❜str❛❝t
❚❤❡ ❛✐♠ ♦❢ t❤✐s ❞✐ss❡rt❛t✐♦♥ ✐s t♦ ✐♥tr♦❞✉❝❡ t❤❡ ♠❛✐♥ ❝♦♥❝❡♣ts ❛♥❞ r❡s✉❧ts ♦❢ ❤②✲ ♣❡r❜♦❧✐❝ ❣❡♦♠❡tr② ✐♥❝❧✉❞✐♥❣ t❤❡ ♥♦♥✲❡①✐st❡♥❝❡ ♦❢ r❡❝t❛♥❣❧❡s✳ ❚❤✐s st❛t❡♠❡♥t ✐s ♦♥❡ ♦❢ t❤❡ ♠❛♥② ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr② ❛♥❞ ❍②♣❡r❜♦❧✐❝ ❣❡♦♠❡tr② ❢r♦♠ t❤❡ ♥❡❣❛t✐♦♥ ♦❢ t❤❡ ❋✐❢t❤ ❆①✐♦♠ ♦❢ ❊✉❝❧✐❞ ♦r ❛s ✐t ✐s ❦♥♦✇♥✱ t❤❡ ❆①✐♦♠ ♦❢ ♣❛r❛❧❧❡❧ ♦❢ ❊✉❝❧✐❞✳ ■♥ t❤❡ ✜♥❛❧ ♣❛rt ♦❢ t❤✐s ✇♦r❦ ✇❡ s❤❛❧❧ ❝♦✈❡r t❤r❡❡ ♠❛✐♥ ♠♦❞❡❧s ♦❢ ❍②♣❡r❜♦❧✐❝ ●❡♦♠❡tr②✿ ❇❡❧tr❛♠✐✲❑❧❡✐♥✱ P♦✐♥❝❛ré ❉✐s❦ ❛♥❞ t❤❡ P♦✐♥❝❛ré ❍❛❧❢ P❧❛♥❡✳ ❲❡ ❛❧s♦ ❞❡♠♦♥str❛t❡ t❤❛t t❤❡s❡ ♠♦❞❡❧s ❛r❡ ✐s♦♠♦r♣❤✐❝✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ A❡ D♣♦♥t♦s ❞✐st✐♥t♦s s♦❜r❡ ✉♠❛ r❡t❛l ❝♦♠B ❡E ❡♠ ❧❛❞♦s ♦♣♦st♦s ❞❡ l✳ ✶✺
✷✳✷ ➶♥❣✉❧♦s ❛❧t❡r♥♦s ✐♥t❡r♥♦s ❝♦♥❣r✉❡♥t❡s⇒ r❡t❛s ♣❛r❛❧❡❧❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸ D ❡ C ❡stã♦ ♥♦ ♠❡s♠♦ s❡♠✐♣❧❛♥♦ ❧✐♠✐t❛❞♦ ♣♦r t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷✳✹ D ❡ C ❡stã♦ ❡♠ s❡♠✐♣❧❛♥♦s ♦♣♦st♦s ❧✐♠✐t❛❞♦s ♣♦r t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✺ ➶♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❝♦♥❣r✉❡♥t❡s⇒ r❡t❛s ♣❛r❛❧❡❧❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✶ ➶♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✷ ➶♥❣✉❧♦s ❞♦ tr✐â♥❣✉❧♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✺✳✶ ❈♦♥str✉çã♦ ❞❡ r❡t❛s ♥♦ ❙❡♠✐♣❧❛♥♦ ❞❡ P♦✐♥❝❛ré✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✺✳✷ ❘❡t❛s ♣❛r❛❧❡❧❛s à ✉♠❛ r❡t❛ ❞❛❞❛ ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦P ❢♦r❛ ❞❛ r❡t❛ ❞❛❞❛✳ ✸✸
✺✳✸ ❘❡t❛s ♣❛r❛❧❡❧❛s à ✉♠❛ r❡t❛ ❞❛❞❛ ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦P ❢♦r❛ ❞❛ r❡t❛ ❞❛❞❛✳ ✹✶
✺✳✹ ❈♦♥str✉çã♦ ❞❡ r❡t❛s ♥♦ ❉✐s❝♦ ❞❡ P♦✐♥❝❛ré✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✺✳✺ ▼❡❞✐❞❛ ❞❡ â♥❣✉❧♦ ♥♦ ❉✐s❝♦ ❞❡ P♦✐♥❝❛ré✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✺✳✻ ❚r✐â♥❣✉❧♦ ♥♦ ❉✐s❝♦ ❞❡ P♦✐♥❝❛ré ❞❡✜♥✐❞♦ ♣❡❧♦s ♣♦♥t♦sA✱ B ❡ C✳ ✳ ✳ ✳ ✳ ✹✹
✺✳✼ P♦♥t♦s ✐❞❡❛✐s ♥♦ ❉✐s❝♦ ❞❡ P♦✐♥❝❛ré✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✺✳✽ ❘❡t❛s ♣❛r❛❧❡❧❛s ❛ss✐♥tót✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✺✳✾ ❯♠ tr✐â♥❣✉❧♦ ❛ss✐♥tót✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✺✳✶✵ ❊①❡♠♣❧♦s ❞❡ r❡t❛s ♥♦ ❉✐s❝♦ ❞❡ ❇❡❧tr❛♠✐✲❑❧❡✐♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✺✳✶✶ ■s♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ▼♦❞❡❧♦s ❞♦ ❉✐s❝♦ ❞❡ ❇❡❧tr❛♠✐✲❑❧❡✐♥ ❡ ❉✐s❝♦ ❞❡
P♦✐♥❝❛ré✳ ❋♦♥t❡✿ ❱❡♥❡♠❛ ❬✶✱ ♣✳ ✷✾✼❪ ✲ ❆❞❛♣t❛❞♦ ♣❡❧♦ ❛✉t♦r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✺✳✶✷ ■s♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ▼♦❞❡❧♦s ❞♦ ❉✐s❝♦ ❞❡ ❇❡❧tr❛♠✐✲❑❧❡✐♥ ❡ ♦ ❙❡♠✐♣❧❛♥♦
❞❡ P♦✐♥❝❛ré✳ ❋♦♥t❡✿ ❱❡♥❡♠❛ ❬✶✱ ♣✳ ✷✾✾❪ ✲ ❆❞❛♣t❛❞♦ ♣❡❧♦ ❛✉t♦r ✳ ✳ ✳ ✳ ✳ ✺✽ ✺✳✶✸ ■s♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ▼♦❞❡❧♦s ❞♦ ❉✐s❝♦ ❞❡ ❇❡❧tr❛♠✐✲❑❧❡✐♥✱ ❙❡♠✐♣❧❛♥♦ ❞❡
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✵
✷ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ ✶✶
✷✳✶ ❚❡r♠♦s ♥ã♦ ❞❡✜♥✐❞♦s✱ ❞❡✜♥✐çõ❡s ❡ ❛①✐♦♠❛s ❞❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ ✳ ✳ ✳ ✳ ✶✶ ✷✳✷ Pr✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✸ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✷✷
✸✳✶ ❚❡r♠♦s ♥ã♦ ❞❡✜♥✐❞♦s✱ ❞❡✜♥✐çõ❡s ❡ ❛①✐♦♠❛s ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✳ ✳ ✷✷ ✸✳✷ Pr✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✹ ●❡♦♠❡tr✐❛ ❍✐♣❡r❜ó❧✐❝❛ ✷✻
✹✳✶ Pr✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ❣❡♦♠❡tr✐❛ ❤✐♣❡r❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✺ ▼♦❞❡❧♦s ♣❛r❛ ❛ ●❡♦♠❡tr✐❛ ❍✐♣❡r❜ó❧✐❝❛ ✷✽
✺✳✶ ▼♦❞❡❧♦ ❝❛rt❡s✐❛♥♦ ❞♦ ♣❧❛♥♦ ❡✉❝❧✐❞✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✺✳✷ ▼♦❞❡❧♦ ❞♦ ❙❡♠✐♣❧❛♥♦ ❞❡ P♦✐♥❝❛ré ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✺✳✷✳✶ ❘❡t❛s ❡ s❡♠✐♣❧❛♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✺✳✷✳✷ ❉✐stâ♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✺✳✷✳✸ ▼❡❞✐❞❛ ❞❡ â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✺✳✷✳✹ ❱❡r✐✜❝❛çã♦ ❞❛ ✈❛❧✐❞❛❞❡ ❞♦s ❆①✐♦♠❛s ❞❛ ●❡♦♠❡tr✐❛ ❍✐♣❡r❜ó❧✐❝❛ ✳ ✸✷ ✺✳✷✳✺ ❆ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛ ♥♦ ❙❡♠✐♣❧❛♥♦ ❞❡ P♦✐♥❝❛ré ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✺✳✸ ❖ ▼♦❞❡❧♦ ❞♦ ❉✐s❝♦ ❞❡ P♦✐♥❝❛ré ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✺✳✸✳✶ ❘❡t❛s ❡ s❡♠✐♣❧❛♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✺✳✸✳✷ ❉✐stâ♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✺✳✸✳✸ ▼❡❞✐❞❛ ❞❡ â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✺✳✸✳✹ ❆ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛ ♥♦ ❉✐s❝♦ ❞❡ P♦✐♥❝❛ré ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✺✳✹ ❖ ♠♦❞❡❧♦ ❞♦ ❉✐s❝♦ ❞❡ ❇❡❧tr❛♠✐✲❑❧❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✺✳✹✳✶ ❉✐stâ♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✺✳✹✳✷ ▼❡❞✐❞❛ ❞❡ â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✺✳✺ ■s♦♠♦r✜s♠♦s ❡♥tr❡ ♦s ♠♦❞❡❧♦s ❞❛ ●❡♦♠❡tr✐❛ ❍✐♣❡r❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✺✳✺✳✶ ❆ ♣r♦❥❡çã♦ ❡st❡r❡♦❣rá✜❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✺✳✺✳✷ ■s♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ♠♦❞❡❧♦s ❞❡ ❇❡❧tr❛♠✐✲❑❧❡✐♥ ❡ ♦ ❉✐s❝♦ ❞❡
✺✳✺✳✸ ■s♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ♠♦❞❡❧♦s ❞❡ ❇❡❧tr❛♠✐✲❑❧❡✐♥ ❡ ♦ ❙❡♠✐♣❧❛♥♦ ❞❡ P♦✐♥❝❛ré ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✻ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✻✷
✶ ■♥tr♦❞✉çã♦
❊st❛ ❞✐ss❡rt❛çã♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✐♥tr♦❞✉③✐r ♦s ❝♦♥❝❡✐t♦s ❡ ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ❍✐♣❡r❜ó❧✐❝❛✱ ❜❡♠ ❝♦♠♦ ♦ ❡st✉❞♦ ❞❡ s❡✉s ♠♦❞❡❧♦s✳
P❛r❛ ❢❛❧❛r♠♦s ♥❛ ●❡♦♠❡tr✐❛ ❍✐♣❡r❜ó❧✐❝❛✱ ♣r❡❝✐s❛♠♦s ❛♥t❡s ❢❛❧❛r ♥❛ ●❡♦♠❡tr✐❛ ❊✉✲ ❝❧✐❞✐❛♥❛✳ ❆ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ❢♦✐ ❛♣r❡s❡♥t❛❞❛ ♣❛r❛ t♦❞❛ ❛ ❤✉♠❛♥✐❞❛❞❡ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❜r✐❧❤❛♥t❡ ♣♦r ❊✉❝❧✐❞❡s ✭✸✵✵ ❛❈✮✱ ❡♠ s❡✉s tr❡③❡ ❧✐✈r♦s ✐♥t✐t✉❧❛❞♦s ❖s ❊❧❡♠❡♥t♦s ❬✷❪✳
◆❡♥❤✉♠ ♠❛t❡♠át✐❝♦ t❡✈❡ ♣♦r ♦❜❥❡t✐✈♦ ❞❡s❝♦❜r✐r ❛ ●❡♦♠❡tr✐❛ ❍✐♣❡r❜ó❧✐❝❛✳ ❙✉❛ ❞❡s✲ ❝♦❜❡rt❛ ❢♦✐ ♦ r❡s✉❧t❛❞♦ ❞♦ ❡s❢♦rç♦ ❞❡ ♠✉✐t♦s ♠❛t❡♠át✐❝♦s ❡♠ t❡♥t❛r❡♠ ❞❡♠♦♥str❛r q✉❡ ♦ q✉✐♥t♦ ❛①✐♦♠❛ ❞❡ ❊✉❝❧✐❞❡s ✭❆①✐♦♠❛ ❞❛s ♣❛r❛❧❡❧❛s ❡✉❝❧✐❞✐❛♥❛s✮ ♥ã♦ ❡r❛ ❛①✐♦♠❛ ❡ s✐♠ ♣♦❞❡r✐❛ s❡r ❞❡♠♦♥str❛❞♦ ❛ ♣❛rt✐r ❞❡ ♦✉tr♦s ❛①✐♦♠❛s✳ ❊ss❡s ♠❛t❡♠át✐❝♦s ❛❝r❡❞✐t❛✈❛♠ q✉❡ ♦ q✉✐♥t♦ ❆①✐♦♠❛ ❞❡ ❊✉❝❧✐❞❡s ❡r❛ ✉♠ t❡♦r❡♠❛✳ ❉❛ ♦❜s❡ssã♦ ♣♦r ❡ss❛ ❞❡♠♦♥s✲ tr❛çã♦✱ ❝✉❥♦ ✜♥❛❧ ❢♦✐ ❢r✉str❛♥t❡✱ t❡✈❡ ❝♦♠♦ ❝♦♠♣❡♥s❛çã♦ ❛ ❞❡s❝♦❜❡rt❛ ❞❡ ✉♠❛ ♦✉tr❛ ❣❡♦♠❡tr✐❛✱ ❛ ❝❤❛♠❛❞❛ ●❡♦♠❡tr✐❛ ❍✐♣❡r❜ó❧✐❝❛✳ ❖s ♠ér✐t♦s ❞❡st❛ ❞❡s❝♦❜❡rt❛ ❞❡✈❡♠✲s❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦s ♠❛t❡♠át✐❝♦s ◆✐❝♦❧á✐ ■✈á♥♦✈✐❝❤ ▲♦❜❛❝❤❡✈s❦② ✭✶✼✾✷✲✶✽✺✻✮ ❡ ❏á♥♦s ❇♦❧②❛✐ ✭✶✽✵✷✲✶✽✻✵✮✳
❉✐✈✐❞✐♠♦s ❡st❡ tr❛❜❛❧❤♦ ❡♠ q✉❛tr♦ ♣❛rt❡s✳
◆❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❛❜♦r❞❛r❡♠♦s ❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛✱ q✉❡ ❝♦♥st✐t✉✐ ♦ ♥ú❝❧❡♦ ❝♦✲ ♠✉♠ ♣❛r❛ ❛s ❞✉❛s ❣❡♦♠❡tr✐❛s✱ ❛♣r❡s❡♥t❛♥❞♦ ♦s s❡✉s ❛①✐♦♠❛s✱ ❞❡✜♥✐çõ❡s✱ ❛❧é♠ ❞❡ s❡✉s ♣r✐♥❝✐♣❛✐s t❡♦r❡♠❛s ❡ ❝♦r♦❧ár✐♦s✳
◆❛ s❡❣✉♥❞❛ ♣❛rt❡ tr❛t❛r❡♠♦s ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ❡♥✉♥❝✐❛♥❞♦ ♦ ❛①✐♦♠❛ ❞❛s ♣❛r❛❧❡❧❛s✳ ❉✐s❝✉t✐r❡♠♦s t❛♠❜é♠ ♦s ♣r✐♥❝✐♣❛✐s t❡♦r❡♠❛s ❡ ♦s t❡♦r❡♠❛s q✉❡ sã♦ ❡q✉✐✈❛✲ ❧❡♥t❡s ❛♦ ❛①✐♦♠❛ ❞❛s ♣❛r❛❧❡❧❛s ❡✉❝❧✐❞✐❛♥❛s✳ ❉❡♠♦♥str❛r❡♠♦s t❛♠❜é♠ ♦ ❚❡♦r❡♠❛ q✉❡ é ❛ ❢r♦♥t❡✐r❛ q✉❡ s❡♣❛r❛ ❛s ●❡♦♠❡tr✐❛s ◆❡✉tr❛✱ ❊✉❝❧✐❞✐❛♥❛ ❡ ❍✐♣❡r❜ó❧✐❝❛✳
◆❛ t❡r❝❡✐r❛ ♣❛rt❡ ❛ ●❡♦♠❡tr✐❛ ❍✐♣❡r❜ó❧✐❝❛ é ❛♣r❡s❡♥t❛❞❛ ❝♦♠ ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ sã♦ ❞✐r❡t❛♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡s ❞♦ ❆①✐♦♠❛ ❞❛s ♣❛r❛❧❡❧❛s ❤✐♣❡r❜ó❧✐❝❛s✳ ❖ ♣r✐♥❝✐♣❛❧ ❞❡❧❡s é ♦ ❞❛ ♥ã♦ ❡①✐stê♥❝✐❛ ❞❡ r❡tâ♥❣✉❧♦s✱ ♦✉ ❛✐♥❞❛✱ q✉❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❤✐♣❡r❜ó❧✐❝♦ é s❡♠♣r❡ ♠❡♥♦r ❞♦ q✉❡ ✶✽✵✳
❊ ✜♥❛❧♠❡♥t❡✱ ♥❛ q✉❛rt❛ ♣❛rt❡ ❞✐s❝✉t✐r❡♠♦s ♦s três ♠♦❞❡❧♦s ❞❛ ●❡♦♠❡tr✐❛ ❍✐♣❡r❜ó✲ ❧✐❝❛✱ sã♦ ❡❧❡s✱ ♦ ❙❡♠✐♣❧❛♥♦ ❞❡ P♦✐♥❝❛ré✱ ♦ ❉✐s❝♦ ❞❡ P♦✐♥❝❛ré ❡ ♦ ❉✐s❝♦ ❞❡ ❇❡❧tr❛♠✐✲❑❧❡✐♥✳ ❊♠ t♦❞♦s ♦s ♠♦❞❡❧♦s ❡st✉❞❛❞♦s ❢❛r❡♠♦s t❛♠❜é♠ ❛ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛ ❝♦rr❡s♣♦♥❞❡♥t❡✳ P♦r ✜♠✱ ❝♦♥str✉✐r❡♠♦s ♦s ✐s♦♠♦r✜s♠♦s ❡♥tr❡ ♦s três ♠♦❞❡❧♦s✳
✷ ●❡♦♠❡tr✐❛ ◆❡✉tr❛
◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s q✉❡ ❝❛r❛❝t❡r✐③❛♠ ❛ ❝❤❛♠❛❞❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ ♦✉ ❆❜s♦❧✉t❛✳ ❖ t❡①t♦ ❞❡ ❱❡♥❡♠❛ ❬✶❪ s❡r✈✐✉ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ♣r✐♥❝✐♣❛❧ ♣❛r❛ ❛s ❞❡✜♥✐çõ❡s ❡ ❡♥✉♥❝✐❛❞♦s ❛q✉✐ ❡①♣♦st♦s✳
❯♠ s✐st❡♠❛ ❛①✐♦♠át✐❝♦ é ❝♦♠♣♦st♦ ♣♦r✿
• ❚❡r♠♦s ♥ã♦ ❞❡✜♥✐❞♦s ❡ r❡❧❛çõ❡s ❞❡♥♦♠✐♥❛❞♦s ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s✳
• ❉❡✜♥✐çõ❡s ❜❛s❡❛❞❛s ♥♦s ❝♦♥❝❡✐t♦s ♥ã♦ ❞❡✜♥✐❞♦s ♦✉ ♣r❡✈✐❛♠❡♥t❡ ❞❡✜♥✐❞♦s ❡ q✉❡ ♣♦❞❡♠ s❡r ✐♥tr♦❞✉③✐❞♦s ❛ss✐♠ q✉❡ ♥❡❝❡ssár✐♦s✳
• ❆①✐♦♠❛s ❜❛s❡❛❞♦s ♥♦s ❝♦♥❝❡✐t♦s ♥ã♦ ❞❡✜♥✐❞♦s ♦✉ ♣r❡✈✐❛♠❡♥t❡ ❞❡✜♥✐❞♦s ❡ q✉❡ ❛t✉❛♠ ❝♦♠♦ r❡str✐çõ❡s✳
• ▲❡♠❛s✱ ♣r♦♣♦s✐çõ❡s ❡ t❡♦r❡♠❛s q✉❡ sã♦ ❛✜r♠❛çõ❡s ✈❡r❞❛❞❡✐r❛s ❞❡❞✉③✐❞❛s ❛ ♣❛rt✐r ❞♦s ❛①✐♦♠❛s✳
• ❆s ❛✜r♠❛çõ❡s ✈❡r❞❛❞❡✐r❛s ❞❡❞✉③✐❞❛s ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛s ❞❡ t❡♦r❡♠❛s sã♦ ❞❡♥♦✲ ♠✐♥❛❞❛s ❝♦r♦❧ár✐♦s✳
❯♠ s✐st❡♠❛ ❛①✐♦♠át✐❝♦ t❡♠ ♣♦r ♣r♦♣r✐❡❞❛❞❡s ❞❡s❡❥á✈❡✐s✿ • ❙❡r ❝♦♥s✐st❡♥t❡✱ ✐st♦ é✱ ♥ã♦ ❞❡✈❡ ❝♦♥t❡r ❝♦♥tr❛❞✐çõ❡s✳
• ❙❡r ❝♦♠♣❧❡t♦✱ ♦✉ s❡❥❛✱ t♦❞♦s ♦s t❡♦r❡♠❛s ❞❡✈❡♠ s❡r ❞❡r✐✈❛❞♦s ❛ ♣❛rt✐r ❞♦ s✐st❡♠❛ ❞❡ ❛①✐♦♠❛s✳
• ❙❡r ✐♥❞❡♣❡♥❞❡♥t❡✱ ✐st♦ é✱ ♥❡♥❤✉♠ ❛①✐♦♠❛ ♣♦❞❡ s❡r ♣r♦✈❛❞♦ ❛ ♣❛rt✐r ❞❡ ♦✉tr♦s✳
✷✳✶ ❚❡r♠♦s ♥ã♦ ❞❡✜♥✐❞♦s✱ ❞❡✜♥✐çõ❡s ❡ ❛①✐♦♠❛s ❞❛ ●❡✲
♦♠❡tr✐❛ ◆❡✉tr❛
❊①✐st❡♠ ❞♦✐s t❡r♠♦s ♥ã♦ ❞❡✜♥✐❞♦s✿ ♣♦♥t♦ ❡ r❡t❛✳ ❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❢❛③ ✉s♦ ❞❛ ❧✐♥❣✉❛❣❡♠ ❡ ❛ ♥♦t❛çã♦ ❞❛ ❚❡♦r✐❛ ❞❡ ❈♦♥❥✉♥t♦s ❡ ✉t✐❧✐③❛ ♦s r❡s✉❧t❛❞♦s ❞❛ ▲ó❣✐❝❛ ❈❧áss✐❝❛✳ ❚❛♠❜é♠ ❝♦♥s✐❞❡r❛r❡♠♦s ✈á❧✐❞♦s t♦❞♦s ♦s r❡s✉❧t❛❞♦s ❞♦s ♥ú♠❡r♦s r❡❛✐s R ❝♦♠♦ ❝♦r♣♦ ♦r❞❡♥❛❞♦ ❡ ❝♦♠♣❧❡t♦✳
❚❡r♠♦s ♥ã♦ ❞❡✜♥✐❞♦s✱ ❞❡✜♥✐çõ❡s ❡ ❛①✐♦♠❛s ❞❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ ✶✷
❆①✐♦♠❛ ✷✳✶ ✭❞❛ ❡①✐stê♥❝✐❛✮✳ ❊①✐st❡♠ ♣❡❧♦ ♠❡♥♦s ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s✳
❖s ♣♦♥t♦s s❡rã♦ ❞❡♥♦t❛❞♦s ♣♦r ❧❡tr❛s ♠❛✐ús❝✉❧❛s ❞♦ ❛❧❢❛❜❡t♦ ❧❛t✐♥♦✱ A✱ B✱ C✱ . . .
❆①✐♦♠❛ ✷✳✷ ✭❞♦ ❝♦♥❥✉♥t♦✮✳ ❈❛❞❛ r❡t❛ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s✳
❆s r❡t❛s s❡rã♦ ❞❡♥♦t❛❞❛s ✉t✐❧✐③❛♥❞♦✲s❡ ❧❡tr❛s ♠✐♥ús❝✉❧❛s ❞♦ ❛❧❢❛❜❡t♦ ❧❛t✐♥♦✱ r✱s✱t✱. . .
❆①✐♦♠❛ ✷✳✸ ✭❞❛ ✐♥❝✐❞ê♥❝✐❛✮✳ P❛r❛ ❝❛❞❛ ♣❛r ❞❡ ♣♦♥t♦s ❞✐st✐♥t♦s P ❡ Q✱ ❡①✐st❡ ❡①❛t❛✲
♠❡♥t❡ ✉♠❛ r❡t❛ r ❞❡ t❛❧ ♠♦❞♦ q✉❡ ❛♠❜♦s ♦s ♣♦♥t♦s P ❡ Q ♣❡rt❡♥❝❡♠ ❛ r✳
❉❡♥♦t❛✲s❡ t❛♠❜é♠ ♣♦r ←→P Q ❛ r❡t❛ q✉❡ é ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦s ♣♦♥t♦sP ❡Q✳
❆①✐♦♠❛ ✷✳✹ ✭❞❛ ❞✐stâ♥❝✐❛✮✳ P❛r❛ ❝❛❞❛ ♣❛r ❞❡ ♣♦♥t♦s P ❡ Q✱ ❛ ❞✐stâ♥❝✐❛ ❞❡ P ❛ Q✱
❞❡♥♦t❛❞❛ ♣♦r d(P, Q)✱ é ✉♠ ♥ú♠❡r♦ r❡❛❧ ♥ã♦ ♥❡❣❛t✐✈♦ ❡ ❞❡t❡r♠✐♥❛❞♦ ✉♥✐❝❛♠❡♥t❡ ♣♦r
P ❡ Q✳
❉❡✜♥✐çã♦ ✷✳✶✳ ❚rês ♣♦♥t♦s ❞✐st✐♥t♦s P✱ Q ❡R sã♦ ❞✐t♦s ❝♦❧✐♥❡❛r❡s s❡ ❡①✐st❡ ✉♠❛ r❡t❛ r t❛❧ q✉❡ P✱ Q ❡ R ♣❡rt❡♥❝❡♠ ❛ r✳ ❈❛s♦ ❝♦♥trár✐♦✱ ♦s ♣♦♥t♦s sã♦ ❞✐t♦s ♥ã♦ ❝♦❧✐♥❡❛r❡s✳
❉❡✜♥✐çã♦ ✷✳✷✳ ❙❡❥❛♠ P✱ Q ❡ R ♣♦♥t♦s ❞✐st✐♥t♦s✳ ❖ ♣♦♥t♦ Q é ❞✐t♦ ❡st❛r ❡♥tr❡ ♦s
♣♦♥t♦s P ❡ R s❡✿
✭✐✮ P✱ Q ❡ R sã♦ ❝♦❧✐♥❡❛r❡s✳
✭✐✐✮ d(P, Q) +d(Q, R) = d(P, R)✳
❉❡♥♦t❛✲s❡ ♣♦rP∗Q∗Rs❡ ♦ ♣♦♥t♦Q❡stá ❡♥tr❡P ❡R✳ ➱ ✐♠❡❞✐❛t♦ q✉❡ s❡ P∗Q∗R
❡♥tã♦ R∗Q∗P✳
❉❡✜♥✐çã♦ ✷✳✸✳ ❉❛❞♦s P ❡ Q ♣♦♥t♦s ❞✐st✐♥t♦s✳ ❖ ❝♦♥❥✉♥t♦ P Q ={P, Q} ∪ {X | P ∗
X∗Q} é ❞❡♥♦♠✐♥❛❞♦ ❞❡ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ❧✐❣❛ ♦s ♣♦♥t♦s P ❡ Q✳
◆♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ P Q♦s ♣♦♥t♦sP ❡Q s❡rã♦ ❞❡♥♦♠✐♥❛❞♦s ❡①tr❡♠♦s ❞♦ s❡❣♠❡♥t♦✳
❉❡✜♥✐çã♦ ✷✳✹✳ ❉♦✐s s❡❣♠❡♥t♦s P Q ❡ RS sã♦ ❞✐t♦s s❡❣♠❡♥t♦s ❝♦♥❣r✉❡♥t❡s s❡ d(P, Q) =d(R, S)✳
❉❡♥♦t❛✲s❡ ♣♦rP Q∼=RS ❛ ❝♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ ♦s s❡❣♠❡♥t♦s P Q❡ RS✳
❆①✐♦♠❛ ✷✳✺ ✭❞❛ ré❣✉❛✮✳ P❛r❛ ❝❛❞❛ r❡t❛ r✱ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ❜✐❥❡t♦r❛ f : r → R✱ ❝♦♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r ❞♦✐s ♣♦♥t♦s P ❡ Q ∈ r✱ t❡♠♦s d(P, Q) =
|f(Q)−f(P)|✳
◗✉❛❧q✉❡r ❢✉♥çã♦ q✉❡ s❛t✐s❢❛ç❛ ♦ ❆①✐♦♠❛ ✷✳✺ s❡rá ❞❡♥♦♠✐♥❛❞❛ ❞❡ ❢✉♥çã♦ ❝♦♦r❞❡♥❛❞❛ ♣❛r❛ r✳
❉❡✜♥✐çã♦ ✷✳✺✳ ❙❡ r é ✉♠❛ r❡t❛✱ ♦s ❧❛❞♦s ❞❡rsã♦ ❞♦✐s ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s ♥ã♦ ✈❛③✐♦s
❚❡r♠♦s ♥ã♦ ❞❡✜♥✐❞♦s✱ ❞❡✜♥✐çõ❡s ❡ ❛①✐♦♠❛s ❞❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ ✶✸
❉❡✜♥✐çã♦ ✷✳✻✳ ❯♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s S é ❞✐t♦ ❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦ s❡ ♣❛r❛ t♦❞♦ ♣❛r
❞❡ ♣♦♥t♦s A ❡ B ❡♠ S✱ ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ AB ❡stá ✐♥t❡✐r❛♠❡♥t❡ ❝♦♥t✐❞♦ ❡♠ S✳
❆①✐♦♠❛ ✷✳✻ ✭❞❛ s❡♣❛r❛çã♦ ❞♦ ♣❧❛♥♦✮✳ P❛r❛ t♦❞❛ r❡t❛ r✱ ♦s ♣♦♥t♦s q✉❡ ♥ã♦ ❡stã♦ ❡♠ r ❞❡✜♥❡♠ ❞♦✐s ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s✱ ♥ã♦ ✈❛③✐♦s✱ Hr+ ❡ Hr− ❝❤❛♠❛❞♦s ❞❡ s❡♠✐♣❧❛♥♦s
❧✐♠✐t❛❞♦s ♣♦r r t❛✐s q✉❡ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s✿
✭✐✮ Hr+ ❡ H−
r sã♦ ❝♦♥✈❡①♦s✳
✭✐✐✮ ❙❡ P ∈Hr+ ❡ Q∈H−
r ❡♥tã♦ P Q∩r 6=∅✳
❉❡✜♥✐çã♦ ✷✳✼✳ ❉❛❞♦sP ❡Q♣♦♥t♦s ❞✐st✐♥t♦s✳ ❖ ❝♦♥❥✉♥t♦−→P Q=P Q∪{X |P∗Q∗X} é ❞❡♥♦♠✐♥❛❞♦ ❞❡ s❡♠✐rr❡t❛ ❞❡ ♦r✐❣❡♠ P ♣❛ss❛♥❞♦ ♣♦r Q✳
❉❡✜♥✐çã♦ ✷✳✽✳ ❙❡❥❛♠ ❞✉❛s s❡♠✐rr❡t❛s −→OA ❡ −−→OB ❞✐st✐♥t❛s ❞❡ ♠❡s♠❛ ♦r✐❣❡♠ ❖✳ ❙❡ ♦s
♣♦♥t♦s A✱ O ❡ B sã♦ ❝♦❧✐♥❡❛r❡s t❛✐s q✉❡A∗O∗B ❞✐③✲s❡ q✉❡−→OA ❡ −−→OB sã♦ s❡♠✐rr❡t❛s
♦♣♦st❛s✳ ❈❛s♦ ♦s três ♣♦♥t♦s ♥ã♦ s❡❥❛♠ ❝♦❧✐♥❡❛r❡s ❞✐③✲s❡ q✉❡ ❛s s❡♠✐rr❡t❛s sã♦ ♥ã♦ ♦♣♦st❛s✳
❉❡✜♥✐çã♦ ✷✳✾✳ ❆ ✉♥✐ã♦ ❞❡ ❞✉❛s s❡♠✐rr❡t❛s −→OP ❡ −→OQ ♥ã♦ ♦♣♦st❛s✱ q✉❡ tê♠ ♦ ♠❡s♠♦
♣♦♥t♦ ✐♥✐❝✐❛❧ O✱ é ❝❤❛♠❛❞♦ ❞❡ â♥❣✉❧♦✳
❖ â♥❣✉❧♦ é ❞❡♥♦t❛❞♦ ♣♦r ∠P OQ ♦✉ ∠QOP✳ ❖ ♣♦♥t♦ O é ❝❤❛♠❛❞♦ ✈ért✐❝❡ ❞♦
â♥❣✉❧♦ ❡ ❛s s❡♠✐rr❡t❛s sã♦ ❝❤❛♠❛❞❛s ❞❡ ❧❛❞♦s ❞♦ â♥❣✉❧♦✳
❉❡✜♥✐çã♦ ✷✳✶✵✳ ❙❡❥❛♠ três ♣♦♥t♦s O✱ P ❡Q t❛✐s q✉❡ ❛s s❡♠✐rr❡t❛s−→OP ❡ −→OQsã♦ ♥ã♦
♦♣♦st❛s✳ ❖ ✐♥t❡r✐♦r ❞♦ â♥❣✉❧♦ ∠P OQ é ❞❛❞♦ ♣❡❧❛ ✐♥t❡rs❡çã♦ ❞♦ s❡♠✐♣❧❛♥♦ ❧✐♠✐t❛❞♦
♣❡❧❛ r❡t❛ ←→OP q✉❡ ❝♦♥té♠ ♦ ♣♦♥t♦Q❝♦♠ ♦ s❡♠✐♣❧❛♥♦ ❧✐♠✐t❛❞♦ ♣❡❧❛ r❡t❛←→OQq✉❡ ❝♦♥té♠
♦ ♣♦♥t♦ P✳
❉❡✜♥✐çã♦ ✷✳✶✶✳ ❆ s❡♠✐rr❡t❛ −→OQ❡stá ❡♥tr❡ ❛s s❡♠✐rr❡t❛s −→OP ❡−→OR s❡ ♦ ♣♦♥t♦Q❡stá
♥♦ ✐♥t❡r✐♦r ❞♦ â♥❣✉❧♦ ∠P OR✳
❆①✐♦♠❛ ✷✳✼ ✭❞♦ tr❛♥s❢❡r✐❞♦r✮✳ P❛r❛ t♦❞♦ â♥❣✉❧♦ ∠P OQ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ ❞❡♥♦✲
t❛❞♦ ♣♦r m(∠P OQ) t❛❧ q✉❡✿
✭✐✮ 06m(∠P OQ)<180✳
✭✐✐✮ m(∠P OQ) = 0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ −→OP =−→OQ✳
✭✐✐✐✮ ✭❈♦♥str✉çã♦ ❞♦ â♥❣✉❧♦✮ P❛r❛ ❝❛❞❛ α ∈ R✱ 0 6 α < 180✱ ❡ ♣❛r❛ ❝❛❞❛ ✉♠ ❞♦s
s❡♠✐♣❧❛♥♦s H ❧✐♠✐t❛❞♦s ♣❡❧❛ r❡t❛ ←→OP ❡①✐st❡ ✉♠❛ ú♥✐❝❛ s❡♠✐rr❡t❛−→OQ❝♦♠ Q∈H
t❛❧ q✉❡ m(∠P OQ) = α✳
❆①✐♦♠❛ ✷✳✽ ✭❛❞✐çã♦ ❞❡ â♥❣✉❧♦s✮✳ ❙❡ ❛ s❡♠✐rr❡t❛ −→OQ ❡stá ❡♥tr❡ ❛s s❡♠✐rr❡t❛s −→OP ❡
−→
OR ❡♥tã♦
Pr✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ ✶✹
❉❡✜♥✐çã♦ ✷✳✶✷✳ ❉♦✐s â♥❣✉❧♦s ∠P OQ ❡ ∠RST sã♦ ❞✐t♦s â♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s s❡ m(∠P OQ) =m(∠RST)✳
❉❡♥♦t❛✲s❡ ♣♦r ∠P OQ∼=∠RST ❛ ❝♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ ♦s â♥❣✉❧♦s ∠P OQ❡ ∠RST✳
❉❡✜♥✐çã♦ ✷✳✶✸✳ ❙❡❥❛♠ três ♣♦♥t♦s✱ A✱ B ❡ C ♥ã♦ ❝♦❧✐♥❡❛r❡s✳ ❖ tr✐â♥❣✉❧♦ ∆ABC é
❛ ✉♥✐ã♦ ❞♦s três s❡❣♠❡♥t♦s AB✱ BC ❡ AC✱ ✐st♦ é
∆ABC =AB∪BC∪AC.
◆♦ tr✐â♥❣✉❧♦∆ABC ♦s ♣♦♥t♦sA✱B ❡Cs❡rã♦ ❞❡♥♦♠✐♥❛❞♦s ✈ért✐❝❡s✱ ♦s s❡❣♠❡♥t♦s AB✱BC ❡AC ❞❡ ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦∆ABC ❡ ♦s â♥❣✉❧♦s∠CAB✱∠ABC ❡∠BCA ❞❡
â♥❣✉❧♦s ✐♥t❡r♥♦s✳
❉❡✜♥✐çã♦ ✷✳✶✹✳ ❉♦✐s tr✐â♥❣✉❧♦s ∆ABC ❡ ∆P QR sã♦ ❞✐t♦s ❝♦♥❣r✉❡♥t❡s s❡ ❡①✐st❡
✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ♦s ✈ért✐❝❡s ❞♦ ♣r✐♠❡✐r♦ tr✐â♥❣✉❧♦ ❡ ♦s ✈ért✐❝❡s ❞♦ s❡❣✉♥❞♦ tr✐â♥❣✉❧♦ t❛✐s q✉❡ ♦s ❧❛❞♦s ❡ â♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s sã♦ ❝♦♥❣r✉❡♥t❡s✳
❉❡♥♦t❛✲s❡ ♣♦r ∆ABC ∼= ∆P QR ❛ ❝♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ ♦s tr✐â♥❣✉❧♦s ∆ABC ❡ ∆P QR✳
❆①✐♦♠❛ ✷✳✾ ✭❈♦♥❣r✉ê♥❝✐❛ ▲❛❞♦ ➶♥❣✉❧♦ ▲❛❞♦ ✲ ▲❆▲✮✳ ❙❡ ❞♦✐s tr✐â♥❣✉❧♦s ∆ABC ❡
∆P QR sã♦ t❛✐s q✉❡ AB ∼= P Q✱ ∠ABC ∼= ∠P QR ❡ BC ∼= QR ❡♥tã♦✱ ∆ABC ∼= ∆P QR✳
✷✳✷ Pr✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛
❚❡♦r❡♠❛ ✷✳✶✳ ❙❡l ❡ m sã♦ r❡t❛s ❞✐st✐♥t❛s✱ ♥ã♦ ♣❛r❛❧❡❧❛s✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♣♦♥t♦ P t❛❧ q✉❡ m∩l =P✳
❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ ♣♦r ❤✐♣ót❡s❡ l ❡ m ♥ã♦ sã♦ ♣❛r❛❧❡❧❛s ❡♥tã♦ m∩l 6=∅✳ ❙✉♣♦♥❤❛♠♦s ♣♦r ❛❜s✉r❞♦ q✉❡m∩l ={P, Q}❝♦♠ P 6=Q✳ P❡❧♦ ❆①✐♦♠❛ ✷✳✸ ❛ r❡t❛ q✉❡
♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s P ❡ Q é ú♥✐❝❛✳ ▲♦❣♦✱ l =m ♦ q✉❡ ❝♦♥tr❛❞✐③ ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡ l ❡ m sã♦ ❞✐st✐♥t❛s✳
▲❡♠❛ ✷✳✶✳ ❙❡❥❛♠ l ✉♠❛ r❡t❛ ❡ A ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r s♦❜r❡ l✳ ❙❡❥❛ B ✉♠ ♣♦♥t♦ ♥ã♦
♣❡rt❡♥❝❡♥t❡ ❛ l✳ ❙❡ C é ✉♠ ♣♦♥t♦ s♦❜r❡ ❛ s❡♠✐rr❡t❛ −→AB ❡ C 6=A✱ ❡♥tã♦ B ❡ C ❡stã♦
❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❡ l✳
❉❡♠♦♥str❛çã♦✳ ❉❡✈❡♠♦s ♣r♦✈❛r q✉❡ BC∩l=∅✳ ❊①✐st❡♠ ❞♦✐s ❝❛s♦s ❛ s❡r❡♠ ❝♦♥s✐❞❡✲
r❛❞♦s✿ A∗C∗B ♦✉ A∗B ∗C✳ ❙❡ A∗C ∗B ❡♥tã♦ A ♥ã♦ ❡stá ❡♥tr❡ B ❡ C✱ ❧♦❣♦ A
♥ã♦ ❡stá s♦❜r❡ ♦ s❡❣♠❡♥t♦BC✳ P❡❧♦ ❚❡♦r❡♠❛ ✷✳✶ ❛s r❡t❛s l❡←→AB tê♠ s♦♠❡♥t❡ ♦ ♣♦♥t♦ A ❡♠ ❝♦♠✉♠✳ ▼❛s A ♥ã♦ ❡stá s♦❜r❡ ♦ s❡❣♠❡♥t♦✱ ❧♦❣♦ BC ∩l = ∅✳ ❆♥❛❧♦❣❛♠❡♥t❡✱
s❡ A∗B∗C ❡♥tã♦ B ❡ C ❡stã♦ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❡ l ✭❛r❣✉♠❡♥t♦ ❛♥t❡r✐♦r tr♦❝❛♥❞♦ ♦s
Pr✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ ✶✺
❚❡♦r❡♠❛ ✷✳✷✳ ❙❡❥❛♠l ✉♠❛ r❡t❛ ❡A ❡D♣♦♥t♦s ❞✐st✐♥t♦s s♦❜r❡ l✳ ❙❡B ❡E sã♦ ♣♦♥t♦s
q✉❡ ❡stã♦ ❡♠ ❧❛❞♦s ♦♣♦st♦s ❞❡ l✱ ❡♥tã♦ −→AB∩−−→DE =∅✳
❉❡♠♦♥str❛çã♦✳ P❡❧♦ ▲❡♠❛ ✷✳✶ ❛s s❡♠✐r❡t❛s −→AB ❡ −−→DE✱ ❡①❝❡t♦ s❡✉s ❡①tr❡♠♦s ❡stã♦ ❡♠
s❡♠✐♣❧❛♥♦s ♦♣♦st♦s ❞❡✜♥✐❞♦s ♣❡❧❛ r❡t❛ ←→AD✳ ❱❡r ✜❣✉r❛ ✷✳✶✳
❋✐❣✉r❛ ✷✳✶✿ A❡D ♣♦♥t♦s ❞✐st✐♥t♦s s♦❜r❡ ✉♠❛ r❡t❛l ❝♦♠B ❡E ❡♠ ❧❛❞♦s ♦♣♦st♦s ❞❡l✳
P❡❧♦ ❛①✐♦♠❛ ✷✳✻ ♦s s❡♠✐♣❧❛♥♦s ❞❡✜♥✐❞♦s ♣♦rlsã♦ ❞✐s❥✉♥t♦s✳ P♦rt❛♥t♦✱ ♦ ú♥✐❝♦ ♣♦♥t♦
q✉❡ ❛s s❡♠✐rr❡t❛s −→AB ❡ −−→DE ♣♦❞❡r✐❛♠ s❡ ✐♥t❡r❝❡♣t❛r s❡r✐❛ ❡♠ s❡✉s ♣♦♥t♦s ❡①tr❡♠♦s✳
▼❛s ✐ss♦ ❝♦♥tr❛r✐❛ ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡ ♦s ♣♦♥t♦s ❡①tr❡♠♦s sã♦ ❞✐st✐♥t♦s✳
❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r é ✉♠ ❡①❡♠♣❧♦ ❞❡ ✉♠ r❡s✉❧t❛❞♦ q✉❡ ❤✐st♦r✐❝❛♠❡♥t❡ ❢♦✐ ❛♣r❡s❡♥✲ t❛❞♦ ❝♦♠♦ ✉♠ ❛①✐♦♠❛✳ ❊st❡ t❡♦r❡♠❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❆①✐♦♠❛ ❞❡ P❛s❝❤✳ Pr♦✈❛r❡♠♦s q✉❡ ❡st❡ r❡s✉❧t❛❞♦ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❆①✐♦♠❛ ✷✳✻ ✭❙❡♣❛r❛çã♦ ❞♦ P❧❛♥♦✮✳ ❉❡ ❢❛t♦✱ ❛ ♣r♦✈❛ r❡✈❡❧❛ q✉❡ ❡st❡ t❡♦r❡♠❛ é ✉♠❛ r❡❛✜r♠❛çã♦ ❞❡ ♣❛rt❡ ❞♦ ❆①✐♦♠❛ ❞❛ ❙❡♣❛r❛çã♦ ❞♦ P❧❛♥♦ ❝♦♠ ✉♠❛ t❡r♠✐♥♦❧♦❣✐❛ ❞✐❢❡r❡♥t❡✳
❚❡♦r❡♠❛ ✷✳✸✳ ❙❡❥❛♠ ∆ABC ✉♠ tr✐â♥❣✉❧♦ ❡ l ✉♠❛ r❡t❛ t❛❧ q✉❡ ♥❡♥❤✉♠ ❞♦s ♣♦♥t♦s A✱ B ❡ C ♣❡rt❡♥ç❛ ❛ l✳ ❙❡ l ✐♥t❡r❝❡♣t❛ ✉♠ ❞♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦✱ ❡♥tã♦ l t❛♠❜é♠
✐♥t❡r❝❡♣t❛ ✉♠ ❞♦s s❡✉s ♦✉tr♦s ❞♦✐s ❧❛❞♦s✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ Hl+ ❡ H−
l ♦s s❡♠✐♣❧❛♥♦s ❞❡t❡r♠✐♥❛❞♦s ♣♦r l✳ ❙❡♠ ♣❡r❞❛ ❞❡
❣❡♥❡r❛❧✐❞❛❞❡✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ l ✐♥t❡r❝❡♣t❛ ♦ ❧❛❞♦AB✳ ❉❡st❛ ❢♦r♠❛✱ ♦s ♣♦♥t♦sA ❡ B ❡stã♦ ❡♠ s❡♠✐♣❧❛♥♦s ♦♣♦st♦s ❞❡t❡r♠✐♥❛❞♦s ♣♦rl✱ ♦✉ s❡❥❛A∈Hl+ ❡B ∈H−
l ✳ ❙❡❣✉❡
q✉❡✱ ♦✉ C ∈ Hl+ ♦✉ C ∈ H−
l ✳ ❙❡ C ∈ H
+
l ❡♥tã♦ CB ∩l 6= ∅✳ ❙❡ C ∈ Hl− ❡♥tã♦
AC∩l 6=∅✳
❉❡✜♥✐çã♦ ✷✳✶✺✳ ❉♦✐s â♥❣✉❧♦s ∠BAD ❡ ∠DAC ❢♦r♠❛♠ ✉♠ ♣❛r ❧✐♥❡❛r s❡ ❛s s❡♠✐r✲
Pr✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ ✶✻
❚❡♦r❡♠❛ ✷✳✹✳ ❙❡❥❛♠ A✱ B✱ C ❡D♣♦♥t♦s ❞✐st✐♥t♦s ♥ã♦ ❝♦❧✐♥❡❛r❡s t❛✐s q✉❡ ♦s ♣♦♥t♦sC
❡ D ♣❡rt❡♥❝❡♠ ❛♦ ♠❡s♠♦ s❡♠✐♣❧❛♥♦ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧❛ r❡t❛ ←→AB✳ ❊♥tã♦✱ m(∠BAD)< m(∠BAC) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ s❡♠✐r❡t❛ −−→AD ❡stá ❡♥tr❡ ❛s s❡♠✐r❡t❛s −→AB ❡ −→AC✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ ♦s ♣♦♥t♦s A✱ B✱ C ❡ D t❛✐s q✉❡ s❛t✐s❢❛③❡♠ ❛ ❤✐♣ót❡s❡ ❞♦ ❡♥✉♥✲
❝✐❛❞♦✳
[⇐]❙✉♣♦♥❤❛♠♦s q✉❡ ❛ s❡♠✐r❡t❛ −−→AD ❡stá ❡♥tr❡ ❛s s❡♠✐r❡t❛s −→AB ❡ −→AC✳ P❡❧♦ ❆①✐♦♠❛
✷✳✽✱
m(∠BAD) +m(∠DAC) = m(∠BAC).
❈♦♠♦ m(∠DAC)>0❡♥tã♦ m(∠BAD)< m(∠BAC)✳
[⇒]P❡❧❛ ❝♦♥tr❛✲♣♦s✐t✐✈❛✱ s✉♣♦♥❤❛♠♦s q✉❡ ❛ s❡♠✐r❡t❛ −−→AD♥ã♦ ❡stá ❡♥tr❡ ❛s s❡♠✐r❡t❛s
−→
AB ❡ −→AC✳ ❙❡ D ∈ −→AC ❡♥tã♦ m(∠BAD) = m(∠BAC)✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ D6∈ −→
AC ❡♥tã♦ C ❡stá ♥♦ ✐♥t❡r✐♦r ❞♦ â♥❣✉❧♦ ∠BAD✳ ▲♦❣♦✱m(∠BAD)> m(∠BAC)✳
P♦rt❛♥t♦✱ m(∠BAD)>m(∠BAC)✳
❚❡♦r❡♠❛ ✷✳✺ ✭❞♦ ♣❛r ❧✐♥❡❛r✮✳ ❙❡ ♦s â♥❣✉❧♦s ∠BAD ❡ ∠DAC ❢♦r♠❛♠ ✉♠ ♣❛r ❧✐♥❡❛r✱
❡♥tã♦
m(∠BAD) +m(∠DAC) = 180.
❉❡♠♦♥str❛çã♦✳ P♦r ❤✐♣ót❡s❡✱ ♦s â♥❣✉❧♦s ∠BAD ❡ ∠DAC ❢♦r♠❛♠ ✉♠ ♣❛r ❧✐♥❡❛r✳
❆ss✐♠✱ ❛s s❡♠✐r❡t❛s −→AB ❡ −→AC sã♦ s❡♠✐rr❡t❛s ♦♣♦st❛s✳ P♦r tr✐❝♦t♦♠✐❛✱ t❡♠♦s q✉❡ m(∠BAD) +m(∠DAC) ♦✉ é ♠❡♥♦r q✉❡ ✶✽✵ ♦✉ é ✐❣✉❛❧ ❛ ✶✽✵ ♦✉ é ♠❛✐♦r q✉❡ ✶✽✵✳
P♦r ❝♦♥tr❛❞✐çã♦✱ s✉♣♦♥❤❛♠♦s q✉❡ m(∠BAD) +m(∠DAC) < 180✳ P❡❧♦ ❆①✐♦♠❛ ✷✳✼
✭✐✐✐✮✱ ❡①✐st❡ ✉♠ ♣♦♥t♦E ♥♦ ♠❡s♠♦ s❡♠✐♣❧❛♥♦ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧❛ r❡t❛←→AC q✉❡ ❝♦♥té♠ ♦
♣♦♥t♦ D✱ t❛❧ q✉❡ m(∠BAE) = m(∠BAD) +m(∠DAC)✳ P❡❧♦ ❚❡♦r❡♠❛ ✷✳✹ ♦ ♣♦♥t♦D
❡stá ♥♦ ✐♥t❡r✐♦r ❞♦ ∠BAE✳ P❡❧♦ ❆①✐♦♠❛ ✷✳✽✱ m(∠BAD) +m(∠DAE) = m(∠BAE)✳
❆ss✐♠✱ m(∠DAE) = m(∠DAC)✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ♣♦♥t♦E ❡stá ♥♦ ✐♥t❡r✐♦r∠(DAC)✳
❆♣❧✐❝❛♥❞♦ ♥♦✈❛♠❡♥t❡ ♦ ❆①✐♦♠❛ ✷✳✽✱ m(∠DAE) +m(∠EAC) =m(∠DAC)✳ P♦rt❛♥t♦✱
m(∠EAC) = 0✱ ♦ q✉❡ ❝♦♥tr❛❞✐③ ♦s ✐t❡♥s ✭✐✮ ❡ ✭✐✐✮ ❞♦ ❆①✐♦♠❛ ✷✳✽✳
P♦r ❝♦♥tr❛❞✐çã♦✱ s✉♣♦♥❤❛♠♦s q✉❡ m(∠BAD) +m(∠DAC) > 180✳ ❙❡❥❛ ♦ ♣♦♥t♦
F ♥♦ ♠❡s♠♦ s❡♠✐♣❧❛♥♦ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧❛ r❡t❛ ←→AC q✉❡ ❝♦♥té♠ ♦ ♣♦♥t♦ D✱ t❛❧ q✉❡ m(∠BAF) =m(∠BAD)+m(∠DAC)−180✳ P❡❧♦ ✐t❡♠ ✭✐✮ ❞♦ ❆①✐♦♠❛ ✷✳✼✱m(∠DAC)<
180✳ ❆ss✐♠✱ m(∠BAF) < m(∠BAD)✳ P❡❧♦ ❚❡♦r❡♠❛ ✷✳✹ ♦ ♣♦♥t♦ F ❡stá ♥♦ ✐♥t❡r✐♦r
❞♦ ∠BAD✳ ▲♦❣♦✱ ♣❡❧♦ ❆①✐♦♠❛ ✷✳✽✱ m(∠BAF) + m(∠F AD) = m(∠BAD)✳ ❆ss✐♠✱
m(∠F AD) = 180−m(∠DAC)✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ♣♦♥t♦ D ❡stá ♥♦ ✐♥t❡r✐♦r ∠(F AC)✳
❆♣❧✐❝❛♥❞♦ ♥♦✈❛♠❡♥t❡ ♦ ❆①✐♦♠❛ ✷✳✽✱m(∠F AD) +m(∠DAC) = m(∠F AC)✳ P♦rt❛♥t♦✱
m(∠BAD) +m(∠DAC)>180✱ ♦ q✉❡ ❝♦♥tr❛❞✐③ ♦ ✐t❡♠ ✭✐✮ ❞♦ ❆①✐♦♠❛ ✷✳✽✳
❉❡✜♥✐çã♦ ✷✳✶✻✳ ❙❡❥❛ ∆ABC ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✳ ❙❡✉s â♥❣✉❧♦s ∠ABC✱ ∠BCA✱
Pr✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ ✶✼
❉❡✜♥✐çã♦ ✷✳✶✼✳ ❙❡❥❛ ∆ABC ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✳ ❯♠ â♥❣✉❧♦ q✉❡ ❢♦r♠❡ ✉♠ ♣❛r
❧✐♥❡❛r ❝♦♠ ✉♠ ❞❛❞♦ â♥❣✉❧♦ ✐♥t❡r♥♦ s❡rá ❞❡♥♦♠✐♥❛❞♦ â♥❣✉❧♦ ❡①t❡r♥♦ ❞♦ tr✐â♥❣✉❧♦✳ ❖❜s❡r✈❡ q✉❡ ♦ tr✐â♥❣✉❧♦ ♣♦ss✉✐ três â♥❣✉❧♦s ✐♥t❡r♥♦s ❡ s❡✐s â♥❣✉❧♦s ❡①t❡r♥♦s✳
❉❡✜♥✐çã♦ ✷✳✶✽✳ ❉♦✐s â♥❣✉❧♦s q✉❛✐sq✉❡r sã♦ ❞✐t♦s s✉♣❧❡♠❡♥t❛r❡s s❡ ❛ s♦♠❛ ❞❛s r❡s✲ ♣❡❝t✐✈❛s ♠❡❞✐❞❛s ❞♦s â♥❣✉❧♦s é ✐❣✉❛❧ ❛ ✶✽✵✳
❖ â♥❣✉❧♦ ✐♥t❡r♥♦ ❢♦r♠❛❞♦ ♣❡❧♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ q✉❡ ❞❡r❛♠ ♦r✐❣❡♠ ❛♦ â♥❣✉❧♦ ❡①t❡r♥♦ é ❝❤❛♠❛❞♦ ❞❡ â♥❣✉❧♦ ❛❞❥❛❝❡♥t❡ ❞♦ â♥❣✉❧♦ ❡①t❡r♥♦✳ ➱ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ ♣❛r ❧✐♥❡❛r✱ ❧♦❣♦ sã♦ s✉♣❧❡♠❡♥t❛r❡s✳
❚❡♦r❡♠❛ ✷✳✻ ✭â♥❣✉❧♦ ❡①t❡r♥♦✮✳ ❆ ♠❡❞✐❞❛ ❞❡ ✉♠ â♥❣✉❧♦ ❡①t❡r✐♦r ❞❡ q✉❛❧q✉❡r tr✐â♥❣✉❧♦ é ❡str✐t❛♠❡♥t❡ ♠❛✐♦r q✉❡ ❛ ♠❡❞✐❞❛ ❞❡ q✉❛❧q✉❡r â♥❣✉❧♦ ✐♥t❡r♥♦ ❛ ❡❧❡ ♥ã♦ ❛❞❥❛❝❡♥t❡✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ ∆ABC ✉♠ tr✐â♥❣✉❧♦ ❡ D ✉♠ ♣♦♥t♦ t❛❧ q✉❡ −−→CD é ♦♣♦st♦ −−→CB✳
❙❡❥❛E ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦AC ❡F ∈−−→BE t❛❧ q✉❡BE ∼=EF✳ ❈♦♠♦ ∠BEA∼=
∠F EC ♣❡❧♦ ❆①✐♦♠❛ ✷✳✾ ♦s tr✐â♥❣✉❧♦s ∆BEA ❡ ∆F EC sã♦ ❝♦♥❣r✉❡♥t❡s ❡ ♣♦rt❛♥t♦
∠F CA ∼= ∠BAC✳ ❙❡ F ❡ B ❡stã♦ ❡♠ ❧❛❞♦s ♦♣♦st♦s ❞❛ r❡t❛ ←→AC ❡ B ❡ D ❡stã♦ ❡♠
❧❛❞♦s ♦♣♦st♦s ❞❡ ←→AC✱ ❡♥tã♦✱ F ❡ D ❡stã♦ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❡ ←→AC✳ ❉♦ ♠❡s♠♦ ♠♦❞♦✱ A ❡ E ❡stã♦ ❡♠ ✉♠ ♠❡s♠♦ ❧❛❞♦ ❞❡ ←→CD ❡ E ❡ F ❡stã♦ ❡♠ ✉♠ ♠❡s♠♦ ❧❛❞♦ ❞❡ ←→CD✳
▲♦❣♦✱ A❡ F ❡stã♦ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❡←→CD✳ ❆ss✐♠✱F ❡stá ♥♦ ✐♥t❡r✐♦r ❞♦ â♥❣✉❧♦∠ACD
❡ m(∠DCA) > m(∠F CA)✳ P♦rt❛♥t♦✱ m(∠DCA) > m(∠BAC)✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡
q✉❡ m(∠DCA)> m(∠ABC) é ❛♥á❧♦❣❛✳
▲❡♠❛ ✷✳✷✳ ❙❡ ∆ABC é ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✱ ❡♥tã♦ m(∠CAB) +m(∠ABC)<180✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ ∆ABC ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✳ ❙❡❥❛ D ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r s♦❜r❡
❛ r❡t❛ ←→AB✱ t❛❧ q✉❡ A ∗ B ∗D✳ ❊♥tã♦✱ m(∠CBD) + m(∠ABC) = 180 ✭❚❡♦r❡♠❛
✷✳✺✮✳ P❡❧♦ ❚❡♦r❡♠❛ ✷✳✻ t❡♠✲s❡ m(∠CAB) < m(∠CBD)✳ ❙❡❣✉❡ q✉❡✱ m(∠CBD) =
m(∠CAB) +p✱ ♦♥❞❡ p∈R✱ ❡ p >0✳ ❉❛í✱ m(∠ABC) +m(∠CAB) +p= 180✳ ❆ss✐♠✱
m(∠ABC) +m(∠CAB) = 180−p✳ P♦rt❛♥t♦✱ m(∠ABC) +m(∠CAB)<180✳
❙❡❥❛∆ABC ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✳ ❉❡♥♦t❛✲s❡ ♣♦rσ(∆ABC) ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s
✐♥t❡r♥♦s ❞♦ tr✐â♥❣✉❧♦ ∆ABC✳
▲❡♠❛ ✷✳✸✳ ❙❡ ∆ABC é ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r ❡ E é ✉♠ ♣♦♥t♦ ♥♦ ✐♥t❡r✐♦r ❞♦ ❧❛❞♦ BC✱ ❡♥tã♦ σ(∆ABE) +σ(∆EAC) = σ(∆ABC) + 180✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ ∆ABC ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r ❡ E ✉♠ ♣♦♥t♦ ♥♦ ✐♥t❡r✐♦r ❞♦ ❧❛❞♦ BC✳ P♦r ❞❡✜♥✐çã♦ ❞❡ s♦♠❛ ❞❡ â♥❣✉❧♦s ✐♥t❡r♥♦s✱ t❡♠✲s❡ σ(∆ABE) +σ(∆EAC) =
m(∠EAB)+m(∠ABE)+m(∠BEA)+m(∠CAE)+m(∠ECA)+m(∠AEC)✳ P♦r ❤✐♣ó✲
t❡s❡✱E❡stá ♥♦ ✐♥t❡r✐♦r ❞♦ ❧❛❞♦BC✱ ❧♦❣♦m(∠EAB)+m(∠CAE) = m(∠BAC)✳ ❚❡♠✲s❡
t❛♠❜é♠ q✉❡ m(∠BEA) + m(∠AEC) = 180 ✭❚❡♦r❡♠❛ ✷✳✺✮✳ P♦rt❛♥t♦✱ σ(∆ABE) +
Pr✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ ✶✽
❉❡✜♥✐çã♦ ✷✳✶✾✳ ❙❡❥❛ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ AB q✉❛❧q✉❡r✳ ❙❡❥❛ M ✉♠ ♣♦♥t♦ q✉❡ ❞❡✲
❝♦♠♣õ❡ ♦ s❡❣♠❡♥t♦ AB ❡♠ ❞♦✐s s❡❣♠❡♥t♦s ❞❡ r❡t❛ t❛✐s q✉❡ ♦s s❡❣♠❡♥t♦s AM ❡ M B
sã♦ ❝♦♥❣r✉❡♥t❡s✳ ◆❡st❡ ❝❛s♦✱ ♦ ♣♦♥t♦ M é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ AB✳
▲❡♠❛ ✷✳✹✳ ❙❡ A✱ B ❡ C sã♦ três ♣♦♥t♦s ♥ã♦ ❝♦❧✐♥❡❛r❡s✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ♣♦♥t♦ D
q✉❡ ♥ã♦ ❡stá s♦❜r❡ ❛ r❡t❛ ←→AB✱ t❛❧ q✉❡ σ(∆ABD) =σ(∆ABC) ❡ ❛ ♠❡❞✐❞❛ ❞❡ ✉♠ ❞♦s
â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ tr✐â♥❣✉❧♦ ∆ABD é ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ 1
2m(∠CAB)✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ A✱ B ❡ C três ♣♦♥t♦s ♥ã♦ ❝♦❧✐♥❡❛r❡s✳ ❙❡❥❛ E ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦
s❡❣♠❡♥t♦ BC✳ ❙❡❥❛D ✉♠ ♣♦♥t♦ s♦❜r❡ ❛ s❡♠✐rr❡t❛−→AE t❛❧ q✉❡A∗E∗D ❡AE ∼=ED✳
❈♦♠♦ ∠AEC ∼= ∠DEB ✭â♥❣✉❧♦s ♦♣♦st♦s ♣❡❧♦ ✈ért✐❝❡✮✱ t❡♠✲s❡ ∆AEC ∼= ∆DEB
✭❝♦♥❣r✉ê♥❝✐❛ ▲❆▲✮ ❡ ❛ss✐♠✱ σ(∆AEC) = σ(∆DEB)✳ ❆♣❧✐❝❛♥❞♦ ♦ ▲❡♠❛ ✷✳✸✱ ❞✉❛s
✈❡③❡s✱ t❡♠✲s❡ q✉❡
σ(∆ABC) = σ(∆ABE) +σ(∆AEC)−180
❡
σ(∆ABD) = σ(∆ABE) +σ(∆DEB)−180.
❉❡st❛ ♠❛♥❡✐r❛✱ s❡❣✉❡ q✉❡ σ(∆ABD) = σ(∆ABC)✳
❱❛♠♦s ❛❣♦r❛ ♠♦str❛r q✉❡✱ ♦✉m(∠DAB)6 1
2m(∠CAB)♦✉m(∠BDA)6 1
2m(CAB)✳
❉❡ ❢❛t♦✱ ♣❡❧♦ ❆①✐♦♠❛ ✷✳✽ t❡♠♦s q✉❡
m(∠BAE) +m(∠EAC) =m(∠BAC)
❆ss✐♠✱ t❡♠♦s ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡ ❛❧❣é❜r✐❝❛s✿ m(∠BAE) 6 1
2m(∠BAC)✱ ♦ q✉❡ ❞❡✲
♠♦♥str❛ ♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ ♦✉ m(∠EAC) 6 1
2m(BAC)✳ ◆❡st❡ ❝❛s♦✱ ❝♦♠♦ ♦ tr✐â♥✲
❣✉❧♦ ∆AEC ∼= ∆DEB t❡♠♦s q✉❡ m(∠EAC) = m(∠EDB)✳ ▲♦❣♦✱ m(∠ADB) =
m(∠EDB) =m(∠EAC)6 1
2m(BAC)✳ ■st♦ ❝♦♠♣❧❡t❛ ❛ ❞❡♠♦♥str❛çã♦✳
❚❡♦r❡♠❛ ✷✳✼ ✭❙❛❝❝❤❡r✐✲▲❡❣❡♥❞r❡✮✳ ◆❛ ❣❡♦♠❡tr✐❛ ♥❡✉tr❛✱ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ❝❛❞❛ tr✐â♥❣✉❧♦ é ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ ✶✽✵✳
❉❡♠♦♥str❛çã♦✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ r❡s✉❧t❛❞♦ s❡rá ❢❡✐t❛ ♣♦r r❡❞✉çã♦ ❛♦ ❛❜s✉r❞♦✳ ❙❡❥❛
∆ABC ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✳ ❙✉♣♦♥❞♦ q✉❡σ(∆ABC)>180✱ s❡❣✉❡ q✉❡σ(∆ABC) = 180 +ǫ✱ ♦♥❞❡ ǫ é ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦✳ ❙❡❥❛ n ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ s✉✜✲
❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ t❛❧ q✉❡ 2nǫ > m(∠CAB)✳ P❡❧♦ ▲❡♠❛ ✷✳✹✱ ❡①✐st❡ ✉♠ tr✐â♥❣✉❧♦ ∆ABC1 t❛❧ q✉❡ σ(∆ABC1) = σ(∆ABC)✱ ❡ ✉♠ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ ∆ABC1✱ t❡♠ ♠❡❞✐❞❛ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ 1
2m(∠CAB)✳ ❆♣❧✐❝❛♥❞♦ ♥♦✈❛♠❡♥t❡ ♦ ▲❡♠❛ ✷✳✹ t❡♠✲s❡ ✉♠ ∆ABC2✱ t❛❧ q✉❡ σ(∆ABC2 = σ(∆ABC)✱ ❡ ✉♠ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ ∆ABC2 t❡♠ ♠❡❞✐❞❛ ✐❣✉❛❧ ♦✉ ♠❡♥♦r q✉❡ 1
4m(∠CAB)✳ ❆♣❧✐❝❛♥❞♦✲s❡ ♦ ▲❡♠❛ ✷✳✹ ✉♠ t♦t❛❧ ❞❡n✈❡③❡s✱
❝❤❡❣❛✲s❡ ❛ ✉♠ ∆ABCn t❛❧ q✉❡ σ(∆ABCn) =σ(∆ABC) = 180 +ǫ✱ ❡ ✉♠ ❞♦s â♥❣✉❧♦s
✐♥t❡r♥♦s ❞♦ ∆ABCn é ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛
1
2nm(∠CAB)< ǫ✳ ▲♦❣♦✱ ❛ s♦♠❛ ❞❛s ♠❡❞✐❞❛s
Pr✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ ✶✾
❉❡✜♥✐çã♦ ✷✳✷✵✳ ❖ â♥❣✉❧♦ ∠ABC é ❞✐t♦ â♥❣✉❧♦ r❡t♦ s❡ m(∠ABC) = 90✳
❉❡✜♥✐çã♦ ✷✳✷✶✳ ❉✉❛s r❡t❛s l ❡ m sã♦ ❞✐t❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s✱ s❡ ❡①✐st❡♠ três ♣♦♥t♦s
❞✐st✐♥t♦s A∈l∩m✱ B ∈l ❡ C ∈m t❛✐s q✉❡ m(∠BAC) = 90✳
❚❡♦r❡♠❛ ✷✳✽✳ ❙❡❥❛♠ m ✉♠❛ r❡t❛ ❡ A ✉♠ ♣♦♥t♦ ♥ã♦ ♣❡rt❡♥❝❡♥t❡ ❛ m✳ ❊①✐st❡ ✉♠❛✱ ❡
s♦♠❡♥t❡ ✉♠❛✱ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ m ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦ A✳
❉❡♠♦♥str❛çã♦✳
✭❛✮ ❊①✐stê♥❝✐❛✿ ❈♦♥s✐❞❡r❡ s♦❜r❡ ❛ r❡t❛ m ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s B ❡ C✳ ❚r❛❝❡ ♦ s❡❣✲
♠❡♥t♦ AB✳ ❙❡ AB é ♣❡r♣❡♥❞✐❝✉❧❛r ❛m✱ t❡r♠✐♥❛✲s❡ ❛ ❝♦♥str✉çã♦✳ ❈❛s♦ ❝♦♥trár✐♦✱
❝♦♥s✐❞❡r❡ ♥♦ s❡♠✐♣❧❛♥♦ ❧✐♠✐t❛❞♦ ♣❡❧❛ r❡t❛ m q✉❡ ♥ã♦ ❝♦♥té♠ A✱ ✉♠ ♣♦♥t♦ X t❛❧
q✉❡ ❛ s❡♠✐rr❡t❛ −−→BX é t❛❧ q✉❡ ♦ â♥❣✉❧♦ ♣♦r ❡❧❛ ❞❡✜♥✐❞♦ ❝♦♠ ❛ s❡♠✐rr❡t❛ −−→BC é
❝♦♥❣r✉❡♥t❡ ❛♦ â♥❣✉❧♦ ∠ABC✱ ♦✉ s❡❥❛✱ ∠XBC ∼= ∠ABC✳ ❙❡❥❛ A′ ∈ −−→BX t❛❧ q✉❡ BA′ ∼= BA✳ ❆ss✐♠✱ ♦ tr✐â♥❣✉❧♦ ∆ABA′ é ✐sós❝❡❧❡s✳ ❈♦♠♦✱ ♣♦r ❝♦♥str✉çã♦✱ ♦ â♥✲ ❣✉❧♦ ∠ABC ∼=∠CBA′✱ ❡♥tã♦−−→BC ✭♦✉ s✉❛ ♦♣♦st❛✮ é ❛ ❜✐ss❡tr✐③ ❞♦ â♥❣✉❧♦∠ABA′✳
❙❡❣✉❡ q✉❡ ←→BC =m é ♣❡r♣❡♥❞✐❝✉❧❛r ❛←→AA′✳
✭❜✮ ❯♥✐❝✐❞❛❞❡✿ ❙❡ ❡①✐st✐ss❡♠ ❞✉❛s r❡t❛s ❞✐st✐♥t❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛ m ♣❛ss❛♥❞♦ ♣❡❧♦
♣♦♥t♦ A ❢♦r♠❛r✲s❡✲✐❛ ✉♠ tr✐â♥❣✉❧♦ ❝♦♠ ❞♦✐s â♥❣✉❧♦s r❡t♦s✱ ♦ q✉❡ ❝♦♥tr❛❞✐③ ♦ ❚❡♦✲
r❡♠❛ ✷✳✼✳
❉❡✜♥✐çã♦ ✷✳✷✷✳ ❖ ❞❡❢❡✐t♦ ❞♦ tr✐â♥❣✉❧♦ ∆ABC é ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ✶✽✵ ❡ ❛ s♦♠❛ ❞♦s
â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡st❡ tr✐â♥❣✉❧♦✳
❉❡♥♦t❛✲s❡ ♦ ❞❡❢❡✐t♦ ❞♦ tr✐â♥❣✉❧♦ ∆ABC ♣♦r✿
δ(∆ABC) = 180−σ(∆ABC).
❉❡✜♥✐çã♦ ✷✳✷✸✳ ❉✉❛s r❡t❛sm ❡ n sã♦ ❞✐t❛s ♣❛r❛❧❡❧❛s s❡ ♥ã♦ ♣♦ss✉❡♠ ♥❡♥❤✉♠ ♣♦♥t♦
❡♠ ❝♦♠✉♠✱ ✐st♦ é✱ m∩n =∅✳
❚❡♦r❡♠❛ ✷✳✾ ✭❚❡♦r❡♠❛ ❞♦s â♥❣✉❧♦s ❛❧t❡r♥♦s ✐♥t❡r♥♦s✮✳ ❙❡ ❞✉❛s r❡t❛s sã♦ ❝♦rt❛❞❛s ♣♦r ✉♠ r❡t❛ tr❛♥s✈❡rs❛❧ q✉❡ ❢❛③ ✉♠ ♣❛r ❞❡ â♥❣✉❧♦s ❛❧t❡r♥♦s ✐♥t❡r♥♦s ❝♦♥❣r✉❡♥t❡s✱ ❡♥tã♦ ❡❧❛s sã♦ ♣❛r❛❧❡❧❛s✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ r ❡ s ❞✉❛s r❡t❛s q✉❡ sã♦ ✐♥t❡r❝❡♣t❛❞❛s ♣❡❧❛ r❡t❛ tr❛♥s✈❡rs❛❧ t
Pr✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ ✷✵
❋✐❣✉r❛ ✷✳✷✿ ➶♥❣✉❧♦s ❛❧t❡r♥♦s ✐♥t❡r♥♦s ❝♦♥❣r✉❡♥t❡s⇒r❡t❛s ♣❛r❛❧❡❧❛s✳
P♦r r❡❞✉çã♦ ❛♦ ❛❜s✉r❞♦✱ ❝♦♥s✐❞❡r❡ r ❡ s ♥ã♦ ♣❛r❛❧❡❧❛s ❡ ❞✐st✐♥t❛s✳ ❆ss✐♠✱ ❡①✐st❡
✉♠ ♣♦♥t♦ D q✉❡ é ❛ ✐♥t❡rs❡çã♦ ❞❡ r ❡ s✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ❝♦♥s✐❞❡r❡ ♦
♣♦♥t♦ C✳ ❙❡D ❡C ❡stã♦ ♥♦ ♠❡s♠♦ s❡♠✐♣❧❛♥♦ ❧✐♠✐t❛❞♦ ♣♦rt✱ ❡♥tã♦ ♦ â♥❣✉❧♦∠A′B′B é ❝♦♥❣r✉♦ ❛♦ â♥❣✉❧♦ ❡①t❡r✐♦r ❞♦ tr✐â♥❣✉❧♦ ∆B′BD ♥♦ ✈ért✐❝❡ B′✳ P♦r ❤✐♣ót❡s❡✱ ♦s â♥❣✉❧♦s∠A′B′B ❡∠CBB′ sã♦ ❝♦♥❣r✉❡♥t❡s✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ â♥❣✉❧♦∠CBB′ é â♥❣✉❧♦ ✐♥t❡r✐♦r ♥ã♦ ❛❞❥❛❝❡♥t❡ ❛♦ â♥❣✉❧♦ ❡①t❡r✐♦r ❞♦ ✈ért✐❝❡ B′✳ ■st♦ ❝♦♥tr❛❞✐③ ♦ ❚❡♦r❡♠❛ ✷✳✻✳ ❱❡r ✜❣✉r❛ ✷✳✸
❋✐❣✉r❛ ✷✳✸✿ D ❡C❡stã♦ ♥♦ ♠❡s♠♦ s❡♠✐♣❧❛♥♦ ❧✐♠✐t❛❞♦ ♣♦rt✳
❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ D ❡ C ❡stã♦ ❡♠ s❡♠✐♣❧❛♥♦s ♦♣♦st♦s ❧✐♠✐t❛❞♦s ♣♦r t t❡♠✲s❡ q✉❡
♦ â♥❣✉❧♦ ∠A′B′B s❡rá â♥❣✉❧♦ ✐♥t❡r✐♦r ♥ã♦ ❛❞❥❛❝❡♥t❡ ❛♦ â♥❣✉❧♦ ❡①t❡r♥♦ ❞♦ tr✐â♥❣✉❧♦
Pr✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ ✷✶
❋✐❣✉r❛ ✷✳✹✿ D ❡C ❡stã♦ ❡♠ s❡♠✐♣❧❛♥♦s ♦♣♦st♦s ❧✐♠✐t❛❞♦s ♣♦rt✳
❈♦r♦❧ár✐♦ ✷✳✶ ✭â♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s✮✳ ❙❡ ❞✉❛s r❡t❛s sã♦ ❝♦rt❛❞❛s ♣♦r ✉♠ r❡t❛ tr❛♥s✈❡rs❛❧ q✉❡ ❢❛③ ✉♠ ♣❛r ❞❡ â♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❝♦♥❣r✉❡♥t❡s✱ ❡♥tã♦ ❡❧❛s sã♦ ♣❛r❛❧❡❧❛s✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ r ❡ s ❞✉❛s r❡t❛s ❝♦rt❛❞❛s ♣♦r ✉♠❛ tr❛♥s✈❡rs❛❧ t ❢♦r♠❛♥❞♦ ✉♠
♣❛r ❞❡ â♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❝♦♥❣r✉❡♥t❡s✳ ❙❡❥❛♠ ♦s ♣♦♥t♦s A✱B✱C✱ A′✱ B′ ✱C′✱B′′ ❝♦♥❢♦r♠❡ ❛ ✜❣✉r❛ ✷✳✺✳
❋✐❣✉r❛ ✷✳✺✿ ➶♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❝♦♥❣r✉❡♥t❡s⇒r❡t❛s ♣❛r❛❧❡❧❛s✳
✸ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛
❆ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ é ♣♦r ❞❡✜♥✐çã♦✱ ❛ ❣❡♦♠❡tr✐❛ q✉❡ s❡ ♦❜té♠ s✉♣♦♥❞♦ t♦❞♦s ♦s ❛①✐♦♠❛s ❞❛ ❣❡♦♠❡tr✐❛ ♥❡✉tr❛ ❡ ♠❛✐s ♦ ❛①✐♦♠❛ ❞❛s ♣❛r❛❧❡❧❛s ❡✉❝❧✐❞✐❛♥❛s✳
✸✳✶ ❚❡r♠♦s ♥ã♦ ❞❡✜♥✐❞♦s✱ ❞❡✜♥✐çõ❡s ❡ ❛①✐♦♠❛s ❞❛ ●❡✲
♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛
❙❡rá ❛❞♦t❛❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t❡r♠♦s ♥ã♦ ❞❡✜♥✐❞♦s✱ ❞❡✜♥✐çõ❡s ❡ ❛①✐♦♠❛s q✉❡ ❞❡✜✲ ♥❡♠ ❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ ✭❆①✐♦♠❛s ✷✳✶ ✕ ✷✳✾✮ ♠❛✐s ♦ ❆①✐♦♠❛ ❞❛s P❛r❛❧❡❧❛s s❡❣✉♥❞♦ ❛ ❢♦r♠✉❧❛çã♦ ❞❡ P❧❛②❢❛✐r✳
❆①✐♦♠❛ ✸✳✶ ✭❆①✐♦♠❛ ❞❛s ♣❛r❛❧❡❧❛s ❞❡ P❧❛②❢❛✐r✮✳ P❛r❛ t♦❞❛ r❡t❛ r ❡ t♦❞♦ ♣♦♥t♦ P q✉❡
♥ã♦ ♣❡rt❡♥❝❡ ❛ r✱ ❡①✐st❡ ❡①❛t❛♠❡♥t❡ ✉♠❛ ú♥✐❝❛ r❡t❛ m✱ t❛❧ q✉❡P ♣❡rt❡♥❝❡ ❛ m ❡ m é
♣❛r❛❧❡❧❛ ❛ r✳
❖ ♠♦t✐✈♦ ❞❡st❛ ❡s❝♦❧❤❛ é t❡r♠♦s ✉♠❛ ♠❛✐♦r ❝❧❛r❡③❛ ❞❛s ❞✐❢❡r❡♥ç❛s ❡♥tr❡ ❛s ❣❡♦♠❡✲ tr✐❛s ❡✉❝❧✐❞✐❛♥❛ ❡ ❤✐♣❡r❜ó❧✐❝❛✳
✸✳✷ Pr✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛
❚♦❞♦s ♦s t❡♦r❡♠❛s ❡ ❝♦r♦❧ár✐♦s ❞❡♠♦♥str❛❞♦s ♥❛ ●❡♦♠❡tr✐❛ ◆❡✉tr❛ sã♦ ❝❧❛r❛♠❡♥t❡ ✈á❧✐❞♦s ♥❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✳
❚❡♦r❡♠❛ ✸✳✶✳ ❙❡ ❛ r❡t❛ m é ♣❛r❛❧❡❧❛ ❛s r❡t❛s r ❡ s✱ ❡♥tã♦ r ❡ s sã♦ ♣❛r❛❧❡❧❛s ♦✉
❝♦✐♥❝✐❞❡♥t❡s✳
❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡r❡s♥ã♦ ❝♦✐♥❝✐❞❡♠ ❡ sã♦ ♣❛r❛❧❡❧❛s ❛ r❡t❛m✳ ❙❡r❡s♥ã♦
❢♦ss❡♠ ♣❛r❛❧❡❧❛s ❡♥tr❡ s✐✱ ❡❧❛s t❡r✐❛♠ ✉♠ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦✱ ❞✐❣❛♠♦s P✳ ❆ss✐♠✱ r ❡s
s❡r✐❛♠ ❞✐st✐♥t❛s ♣❛r❛❧❡❧❛s à r❡t❛ m ♣❛ss❛♥❞♦ ♣♦r P✱ ♦ q✉❡ ❝♦♥tr❛❞✐③ ♦ ❆①✐♦♠❛ ✸✳✶✳
❈♦r♦❧ár✐♦ ✸✳✶ ✭â♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s✮✳ ❙❡ ❞✉❛s r❡t❛s ♣❛r❛❧❡❧❛s sã♦ ❝♦rt❛❞❛s ♣♦r ✉♠❛ r❡t❛ tr❛♥s✈❡rs❛❧✱ ❡♥tã♦✱ ♦s q✉❛tr♦ ♣❛r❡s ❞❡ â♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s sã♦ ❝♦♥❣r✉❡♥t❡s✳
Pr✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✷✸
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠r ❡s❞✉❛s r❡t❛s ♣❛r❛❧❡❧❛s ❡t✉♠❛ r❡t❛ q✉❡ ✐♥t❡r❝❡♣t❛ ❛s r❡t❛sr ❡ s♥♦s ♣♦♥t♦sA❡B✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❥❛ ❛ r❡t❛m✱ ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦A❡ ❢♦r♠❛♥❞♦
❝♦♠ ❛ tr❛♥s✈❡rs❛❧✱ q✉❛tr♦ â♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s ❛♦s â♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❢♦r♠❛❞♦s ♣❡❧❛ r❡t❛ s ❝♦♠ ❛ ♠❡s♠❛ tr❛♥s✈❡rs❛❧ t✳ P❡❧♦ ❝♦r♦❧ár✐♦ ✷✳✶✱ s ❡ m sã♦ ♣❛r❛❧❡❧❛s✳ P❡❧♦
❚❡♦r❡♠❛ ✸✳✶✱r❡msã♦ ❝♦✐♥❝✐❞❡♥t❡s✳ P♦rt❛♥t♦✱r❢♦r♠❛ â♥❣✉❧♦s ❝♦♠ ❛ r❡t❛t❝♦♥❣r✉❡♥t❡s
❛♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❢♦r♠❛❞♦s ❡♥tr❡ ❛s r❡t❛s s ❡ t✳
❋✐❣✉r❛ ✸✳✶✿ ➶♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s✳
❚❡♦r❡♠❛ ✸✳✷✳ ❚♦❞♦ tr✐â♥❣✉❧♦ t❡♠ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ✐❣✉❛❧ ❛ ✶✽✵✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ ∆ABC ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✳ P❡❧♦ ✈ért✐❝❡ C✱ tr❛❝❡ ✉♠❛ r❡t❛ r
♣❛r❛❧❡❧❛ ❛♦ ❧❛❞♦AB✳ ◆✉♠❡r❡ ♦s â♥❣✉❧♦s ❢♦r♠❛❞♦s ❝♦♠ ♦ ✈ért✐❝❡ C✱ ❝♦♠♦ ✐♥❞✐❝❛❞♦ ♥❛
✜❣✉r❛ ✸✳✷✳
Pr✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✷✹
➱ ✐♠❡❞✐❛t♦ q✉❡ m(∠1) +m(∠2) +m(∠3) = 180✳ ❈♦♠♦ AC é tr❛♥s✈❡rs❛❧ às ❞✉❛s
♣❛r❛❧❡❧❛s✱ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞✐r❡t❛ ❞♦ ❝♦r♦❧ár✐♦ ✸✳✶ q✉❡ ∠1 ∼= ∠BAC✳ ❈♦♠♦ BC
é t❛♠❜é♠ tr❛♥s✈❡rs❛❧ às ❞✉❛s ♣❛r❛❧❡❧❛s✱ ❡♥tã♦ ∠3 ∼= ∠ABC✳ P♦rt❛♥t♦✱ m(∠CAB) +
m(∠ACB) +m(∠CBA) =m(∠1) +m(∠2) +m(∠3) = 180✳
❉❡♥♦♠✐♥❛✲s❡ q✉❛❞r✐❧át❡r♦ t♦❞♦ ♣♦❧í❣♦♥♦ ❞❡ ✹ ✈ért✐❝❡s✳ ❖ q✉❛❞r✐❧át❡r♦ P QRS é
❞✐t♦ ❝♦♥✈❡①♦ s❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ✐♥t❡r✐♦r❡s ❞♦ q✉❛❞r✐❧át❡r♦ é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦✳ ❉❡✜♥✐çã♦ ✸✳✶✳ P❛r❛ t♦❞♦ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ P QRS✱ ♦ ❞❡❢❡✐t♦ ❞❡ P QRS é ❛
❞✐❢❡r❡♥ç❛ ❡♥tr❡ ✸✻✵ ❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ q✉❛❞r✐❧át❡r♦✳ ❉❡♥♦t❛✲s❡ ♦ ❞❡❢❡✐t♦ ❞♦ q✉❛❞r✐❧át❡r♦ P QR❙ ♣♦r✿
δ(P QRS) = 360−σ(P QRS).
❉❡✜♥✐çã♦ ✸✳✷✳ ❚♦❞♦ q✉❛❞r✐❧át❡r♦ ❝✉❥♦s â♥❣✉❧♦s ✐♥t❡r♥♦s sã♦ â♥❣✉❧♦s r❡t♦s s❡rá ❞❡♥♦✲ ♠✐♥❛❞♦ ❞❡ r❡tâ♥❣✉❧♦✳
❈♦r♦❧ár✐♦ ✸✳✷✳ ❆ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ r❡tâ♥❣✉❧♦ é ✸✻✵✳
❉❡♠♦♥str❛çã♦✳ ❆ ❝♦♥s❡q✉ê♥❝✐❛ ❞✐r❡t❛ ❞❛ ❞❡✜♥✐çã♦ ❞❡ r❡tâ♥❣✉❧♦ ❥á ❣❛r❛♥t❡ q✉❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s é ✸✻✵✱ ♣♦✐s✱ ❡st❡ q✉❛❞r✐❧át❡r♦ t❡♠ q✉❛tr♦ â♥❣✉❧♦s ✐♥t❡r♥♦s✱ ❝❛❞❛ ✉♠ ✐❣✉❛❧ ❛ ✾✵✳ P♦rt❛♥t♦✱ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦ é ✸✻✵✳
P♦❞❡♠♦s ♠♦str❛r q✉❡ ♦ ❚❡♦r❡♠❛ ✸✳✷ ❡ ♦s ❞❡♠❛✐s ❝♦r♦❧ár✐♦s sã♦ ❡q✉✐✈❛❧❡♥t❡s ❛♦ ❆①✐♦♠❛ ❞❛s ♣❛r❛❧❡❧❛s✳ ❆ss✐♠✱ ❝❛❞❛ ✉♠❛ ❞❛s ❛✜r♠❛çõ❡s ❛❜❛✐①♦ ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞❛ ❝♦♠♦ ❆①✐♦♠❛ ❡ ❛s ❞❡♠❛✐s✱ ✐♥❝❧✉✐♥❞♦ ♦ ❆①✐♦♠❛ ✸✳✶✱ ♣❛ss❛♠ ❛ s❡r t❡♦r❡♠❛s✳ ❙ã♦ ❡❧❛s✿
• ❆ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r é ✶✽✵✳
• ❙❡ três â♥❣✉❧♦s ❞❡ ✉♠ q✉❛❞r✐❧át❡r♦ sã♦ r❡t♦s✱ ❡♥tã♦ ♦ q✉❛rt♦ â♥❣✉❧♦ s❡rá r❡t♦ t❛♠❜é♠✳
• ❊①✐st❡♠ r❡tâ♥❣✉❧♦s✳
P❛r❛ ♠♦str❛♠♦s ❛s ❡q✉✐✈❛❧ê♥❝✐❛s ♥❡❝❡ss✐t❛♠♦s ❞♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
▲❡♠❛ ✸✳✶✳ ❙❡❥❛♠ P Q ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❡Q′ ✉♠ ♣♦♥t♦ t❛❧ q✉❡ ♦ â♥❣✉❧♦ ∠P QQ′ é ✉♠ â♥❣✉❧♦ r❡t♦✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ ǫ >0❡①✐st❡ ✉♠ ♣♦♥t♦ T ∈−−→QQ′ t❛❧ q✉❡ m(∠P T Q)<
ǫ✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛P′ ✉♠ ♣♦♥t♦ ❞♦ ♣❧❛♥♦ t❛❧ q✉❡P′ ♣❡rt❡♥❝❡ ❛♦ ♠❡s♠♦ s❡♠✐♣❧❛♥♦ ❞❡✲ t❡r♠✐♥❛❞♦ ♣❡❧❛ r❡t❛←→P Q❡ ♦ ♣♦♥t♦Q✳ ❈♦♥s✐❞❡r❡♠♦s ❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦sT1, T2, . . . Tn
s♦❜r❡ ❛ s❡♠✐rr❡t❛ −−→QQ′ t❛✐s q✉❡ ✭❛✮ QT1 ∼=P Q✳
